Riding a bicycle on the sidewalks in Bowling Green, OH is illegal; there are signs all over the place indicating this. I guess there was a problem over the years, and they finally decided that this was the solution.
The way I understand it, a bicycle is considered similar to any other vehicle designed for the road, e.g. a car, and is expected to obey all the same laws...including riding in the street where you're supposed to. I also understand that in some places, if you run into a car while riding on the sidewalk it's on you, whereas if you're in the street and they pull out into you it's on them, for insurance reasons. Though I'm sure to them, either way it's *your* fault.
Ride in the street, it's so much easier.