Introduction
I have intended to work out the probability properties of the game MonopolyŽ since I first studied Markov chains in 1985.  It took 16+ years.  I'm glad to say, however, that I did not squander an opportunity to be the first. Unbeknownst to me, Professor Irvin Hentzel had published an analysis in the Saturday Review of the Sciences of April 1973.  Since then there have been several, notably by Ian Stewart, in Scientific American, April 1996.  Here is a quick list of analyses of this problem on the Web (they all worked on 12/30/2001):
Allan Evans http://www.cms.dmu.ac.uk/~ake/monopoly.html
Durango Bill http://www.oocities.org/durangobill/MnplyStats.html
Ian Stewart http://www.math.yorku.ca/Who/Faculty/Steprans/Courses/2042/Monopoly/Stewart2.html
Irvin Hentzel http://www.public.iastate.edu/~hentzel/monopoly/homepage.html
Jim & Mandy http://home.att.net/~dambrosia/programming/games/monopoly/index.html
Truman Collins http://www.tkcs-collins.com/truman/monopoly/monopoly.shtml
Truman Collins's analysis appears to have been crowned the champ, and my work here largely corroborates his.  I was able to advance the ball a little bit by resolving an issue he had left open.  There is still plenty to be done, and difficult issues to wrangle, before the entire Monopoly strategy problem can be said to have been solved.  But I've reached my limit and hereby cheerfully hand off to someone whose grasp of probability and optimization modelling is wider and deeper than mine, and who has more time on his/her hands.  Ph.D. candidates, this is your cue.
Basics
Markov chains (also spelled Markoff chains, but not to be confused with mark-off chains such as Sam's Club or Home Depot) give a method for thinking about the stochastic properties of a finite-state system.  That is, if the probability of moving from state I to state J does not depend on any information from prior states visited, then the system can be modelled as a Markov chain.  If there are N possible states, then the probabilistic properties are entirely captured by a full set of functions Pr(go to state J|currently in state I), 1 <= I,J <= N.  This information can be conveniently arranged into an NxN matrix.
Clearly this applies to the problem of predicting positions in a board game.
To me, the amazing thing about Markov chains is that the convenient matrix tabulation turns out to have analytical teeth.  If the transition probabilities have been stored in a matrix M = {Pr(go to J|begin at I)}, then the probabilities of transitioning from I to J in two moves are simply M*M in normal matrix multiplication, in three moves, M*M*M, and so on.  The second magical fact is that this process converges -- raising M to a sufficiently large power gives N identical rows of transition probabilities; i.e. for any I, Pr(J|I) is the same.  This is called the "ergodic" or "steady state" probability distribution associated with this Markov process.
Recommendation:  I found Cinlar, Introduction to Stochastic Processes, very valuable, especially for those who, like me, have a surer grasp of linear spaces than of probability.
MonopolyŽ
Markov chains help to define a research agenda for studying MonopolyŽ. Specify the transition probabilities, crank up the Markov matrix, and voila!  The steady-state probabilities give the long-run probability of landing on, say, St. James Place, on the next roll (not conditional on where one is sitting). This would seem to be a relatively easy, if tedious, exercise.  From that information, derive optimal strategies.  Unfortunately, there are several complicating factors:
Chance and Community Chest
Ten of 16 Chance cards, and two of 16 Community Chest cards, will send you places.  This is not a conceptual problem, but it does add a layer of tedium in calculating the transition probabilities.  In fact, it is frequently possible to return to the origin square in one roll.  For example, there is an "Advance to St. Charles' Place" Chance card.  Therefore, because you can roll 11 from St. Charles Place and hit Chance, there is a small probability (=(2/36)*(1/16)) that you will be returned to St. Charles Place on the same turn, albeit $200 richer.
Going to Jail
You can go to Jail by hitting the "Go To Jail" square, by getting a "Go To Jail" card in Chance or Community Chest, or by rolling doubles three times in a row.  Modelling the first two is straightforward. 
Doubles
Doubles are particularly problematic.  The "Jail on third doubles" rule appears to negate the attractive features of the Markov chain.  From my home, Marvin Gardens, whether you go to Jail or to Park Place can depend on whether your doubles roll is the third in a row or not, not merely on your position.  An easy fixup beckons:  it might seem that the probability of having rolled doubles twice before the current roll would simply be 1/36 (=(1/6)*(1/6)).  However, this is not correct in general, because sometimes certain sequences of doubles could not have taken you to the square you occupy.  For example, you cannot roll "2" then "4" to reach Pennsylvania Avenue from Water Works, because the "2" would take you straight to Jail.
I confess that I missed this subtlety, and learned of it from Truman Collins' page.  He gives an approximate solution using Monte Carlo methods (I.e. he wrote a program to run thousands of simulated games, yielding estimates of the probabilities).  I give an analytical solution, based on the following reasoning:
Write the vector of steady-state probabilities Pr(go to Jail due to the doubles rule|start at I) as V.  In steady state, the probability of being on any square J after two doubles is the sum of the probabilities of being on each upstream candidate square, I, multiplied by the conditional probability of reaching square J with two doubles (i.e. Z = {Pr(reach J after two doubles|start at I)}, 1 <= I,J <= N).  Clearly Z defines a Markov chain in its own right, and (1/6)(ZV') gives the steady-state probability of going to Jail due to the doubles rule.  But this means that (1/6)(ZV') = V'.
This formulation suggests an iterative method.  I applied such a method, with initial specification of V as a uniform 1/216 (=(1/6)^3).  I used that vector to calculate the implied steady-state probabilities, and from that calculated the implied probabilities of being at State I after two doubles, and substituted that updated estimate for the initial probability vector, and so on.  I did not attempt to prove conditions under which this method would converge, but do report that it did converge, and fairly quickly.  These results are very close to those of Truman Collins (see tab "long-run probs"), each method corroborating the other.
Staying in Jail  
You  can opt to get out of Jail immediately, by paying a $50 fine, or to stay for up to three turns, by not paying the fine. If you get doubles, though, you're out of jail.  In the early game, during the scramble to acquire property, getting out quickly is optimal; later, however, when you're avoiding your rivals' hotels, Jail is a welcome haven.  I give the short-jail probabilities, but for further analysis assume that the long-jail probabilities are the most relevant.
Property Values
Actual property values are a can of worms affected by all of the probabilistic considerations above, parameters such as the number of players, and various strategic decisions made by those players.  Those strategic decision are, in turn, driven by property values.  I leave it to someone else to devise general strategies.  Here, I merely apply the steady-state probabilities to the known value parameters (they are printed on the title deeds) to derive a simple valuation of houses  and hotels on the various properties and color-groups, and the implied payback period for each.  These values are calculated in terms of "rival-rolls," i.e. the expected revenue of a hotel on Illinois Avenue per rival-roll is the amount, on average, that you will collect each time any one or your opponents  rolls the dice.
Caveats:
The actual price of the property is not included in the value calculation for the color-groups, but is included in the valuation of Railroads and Utilities (because these values are driven solely by the number of similar properties owned).
Rents due for Railroads and Utilities are slightly higher when one has been sent there by certain Chance cards.  This factor is ignored.
It is assumed that you have acquired the complete color-group, so as to permit construction of houses and hotels.
Chance and Community Chest each include a card imposing a tax on buildings.  This factor is incorporated into the calculation.
The probabilities used here are the "long-jail" probabilities, reflecting the assumption that when such things matter, the optimal strategy is to hang out in jail as much as possible.  The exact threshold for switching from "short-jail" to "long-jail" is well beyond my ken.
Payback Periods
Payback periods give the expected number of rival-rolls it takes to recoup the cost  of improvements (if  a color-group).  They are calculated in the same units; a payback period of 30 means that after your opponents have rolled the dice 30 times, you can expect to have recouped the costs of erecting the 3 houses, or hotel, or whatever.
Conclusions:
1.  Don't buy the Dark Purples.
2.  Don't buy the Utilities.
3. The Railroads aren't so great, either.
4.  Orange and Dark Blue (Boardwalk, anyway) appear to be the best in terms of payback period, but be aware that the fact there are only two Dark Blues works against their overall benefit.
5.  The rest of the color groups are all roughly similar in terms of payback period, considering all the unmodelled factors.  Greens aren't so great.
6.  Building only one or two houses on a given site is a poor investment.  You need at least three.
7.  Three houses usually appears to be optimal.  The payback periods for three houses, four houses and a hotel are about the same (Although, interestingly, on the cheaper sides of the board, hotels are slightly better than three houses, but on the tonier sides,  vice versa.  If you think of "hotel" as "tenement" this makes perfect sense.).  An additional consideration, not explicitly modelled here, is that adding the houses, or especially upgrading to a hotel, carries significant risk -- if you have to tear down the buildings to pay a debt, you get only half the purchase price back.  Better to keep extra cash on hand.
Comment:
I haven't played  much  Monopoly as an adult, but as kids no one I knew ever played the "optimal" strategy of hiding out in Jail -- my brothers and I would pay our $50 and get back in the game.  I just don't remember Jail being such a common destination.  I suppose that this is because under the "short-jail" strategy, Jail is not quite twice as likely as any other square, but there's still only a 1-in-25 chance of doing time.  I do vividly remember that it was hard to prosper owning the Dark Purples and the Greens.