Okay, so I'm just now getting around to doing something about this. I seem to have solved the problem but now there's a host of new questions:
I lowered the speed of the RAM in the BIOS menu, but it looks like there is no way to do this without lowering the clock speed of the CPU as well (WTF?!). The hit I take is really big too...from 2.54 Ghz to 1.9 Ghz. Am I stuck with this situation until I replace the RAM? Is there some menu option I might be missing?
So far I'm able to play even very stress-filled games without having the system reboot itself. Does this mean that the RAM was definitely the problem? Or could it still be the CPU or PSU (and lowering the clock on everything was what did it)? I'm guessing that I can rule out software issues at this point.
I haven't run the memory test yet since I can't get the system to boot from a CD (even with CD boot enabled) and don't have any floppies around....