Not sure about other places, but in L.A. - bikes can ride on the street, so long as they obey traffic laws. They can also ride on the sidewalks, but they can't ride through crosswalks - they have to stop at the curb and walk them through.
The logic, I assume, is that when drivers turn down a street, they're looking for pedestrians who are generally traveling at one speed, and can stop more-or-less on a dime, if necessary. But too often, bicyclists will shoot through the crosswalk because they have the walk sign and assume they have the right-of-way. So a driver sees no pedestrians, and assumes the coast is clear, but mid-turn, a guy on a bike darts in front of them.
My sense is that, very few bicyclists in L.A. are aware of this law, and I was wondering if other cities had the same thing going.