I've figured it out: it's the Cyborg mouse. I'll be looking through the questions as I remember seeing something about this.
I'm getting a similar issue to this fellow: https://askubuntu.com/questions/75433/ubuntu-11-04-randomly-freezes-for-over-one-minute
Sometimes it comes back to life after a minute or two only to crash again. Alt-tab works, but it does not display the windows-switching animation. It just switches the focus... sometimes. Ctrl-Alt-T works, thankfully, and the terminal stays responsive long enough for me to get in a "sudo reboot now" and type my password.
I'm running a fresh Wubi install on a separate HDD from my Windows install. 64-bit 12.10 12.04.1 LTS now, with an AMD FX chip, 8GB of RAM, and a Radeon HD 3850. My mouse is a Saitek Cyborg R.A.T.5 Mouse, and my keyboard is a stock Acer one that came with a PC I bought a few years ago.