Puzzle Number Two

Since the last puzzle was (rightly) pointed out to be relatively straightforward (though tedious), I thought I would go with a fairly challenging problem for this one. This is actually one of my favourite problems from the probability course I took last year. I should warn you that it is a fairly challenging problem, though, unlike most puzzles, doesn't really take any tricks or subtleties of the language. Instead, solving it will require some skill with calculus and probability theory as well as some relatively straightforward reasoning.

So, without further rambling, the problem is as follows:

You have a floor made up of parallel boards of width D (like a typical hardwood floor) and a uniform metal rod of length L, where L < D. Assuming that when one drops the rod it will land with uniform probability in any position and orientation on one of the floorboards (and, obviously, position is independent from orientation), is it possible to derive an empirical estimate for π through repeatedly dropping the rod and counting the number of intersections with cracks between floorboards? If so, how?