nsvader and Nastyman: I cleaned it out, the problem persisted. I ran Memtest86+ with all four sticks in place, and it resulted in 13,000 write errors. Since I tried running with sticks of RAM from another rig in different slots and still had the random reboots, I doubt that it's specifically the sticks, but I could run it one slot with the same stick to get further suggestion that the mobo is faulty.
kleinkinstein: OCZ 600W Z600SXS.
http://www.ocztechnology.com/ocz-600w-s ... y-eol.html. You and LatiosXT may be correct about the PSU being the culprit. Since it crashes just as easily with sitting on the desktop as it does with running games, I'm more prone to think it is the motherboard. If I were to replace the power supply, I might throw money at it with apparently the best PSU ever according to testing (the
Seasonic X-1250). The power draw of tasks doesn't seem to be linked to the crashes. As wazkatango mentioned, the GPU probably isn't the cause, since it crashed even with a 4850 from another rig. I updated my 4870's video card drivers and CCC to 13.1. I've seen fewer crashes after doing an upgrade install of Win7, and especially after reseating the main 24-pin power connector. The loose connector was directly responsible for power turning off and for it not turning on, confirmed by reseating everything but that connector, so that was at least part of the problem. It could be that the power supply is to blame just because of the connector might have breaks keeping it from staying in place, but that wouldn't explain reboots in the middle of nothing happening.
Nastyman and LatiosXT: In the course of doing this troubleshooting, I've spent several hours staring at the motherboard and didn't notice any unusual caps. I'll mention it if I see anything unusual. Two of the SATA ports were partially dislodged from the board a while back, but I don't think that would cause random shorts.