I decided to try setting up Calibre on my machine to manage my e-books, and rather than use the e-mail sync functionality with my Kindle, I figured I would just sync it via USB. I attached it through another micro-USB cable that I happened to have attached to my computer, but I had weird issues where any file I/O would cause the Kindle USB drive to become unmounted and then remount. This was extremely strange, so I tried removing all software from my computer that might be related to the problem: Virtual CloneDrive, VirtualBox, etc. This didn’t fix it. I updated the USB drivers for my motherboard, to no avail.
Finally, after a bit more Googling, I saw that some people mentioned that they tried plugging the Kindle into a different USB port on their machine. That didn’t work for me, but changing out my cable for the official Kindle cable made it magically work. Very strange!
So for anyone who’s having problems syncing their Kindle via USB, make sure you’re using the official cable (or a high-quality cable, at least). I initially thought it might be a 64-bit Windows 7/Vista problem, because I saw some other reports of problems with it, but at least in my case it turns out that it was the USB cable I was using.