Thanks guys.

I Googled "Award Winning Freeware File Recovery" and downloaded about 17 different programs on top of the ones suggested in this thread. Unfortunately, file recovery programs seem to all deal with undeleting DELETED files, and in this case nothing has been deleted. I expect the problem is a corrupted file allocation table because every computer I put the USB drive in tells me the drive has not been formatted, and asks me if I want to format it. That tells me the problem is with how the USB drive has addressed the files stored on it, not with the files themselves.

Thanks again for everyone's input.