My theory about what is going on is that the drive controller, for some reason, is not able to "wake up" anymore, and so sometimes when you ask for a file it takes a long time to actually read it or write it, but my data did not appear to be damaged--I was able to copy everything off of the drive, it just took multiple tries for some files.
Thanks to a 3 year warranty, I am able to send the drive back for replacement to Western Digital. I'm a little spooked by the failure though, and I am going to replace the drive with a non-Green model for now. I liked how quiet the Caviar Green was, but it wasn't the fastest drive even when it was working.