I've tried this several times now. I tap the buy button, it jumps to the download icon, I type my password, and nothing happens. It absolutely will not download. This is very strange, since I bought an album on the Touch just yesterday and it went fine. It may have nothing to do with free songs at all, but has anyone ever seen this behavior?
EDIT: Okay, I think I figured out the problem. I tried to download the free song from iTunes and was asked to accept new terms of service, and also to update my account balance. Apparently the mobile version of iTunes can't handle one or both of these requests, and aborted the download instead. Once I clicked through these dialog boxes in iTunes on the Mac, I was able to download the song on the Touch.