Just an update. I finally got the system stable, but only because I put in one stick of RAM.
Either of the RAM sticks works and is stable. If I put them in pairs, the system is unstable and just reboots.
BioStar of course tells me it's probably the CPU. I still haven't updated the BIOS but am waiting on an answer from them about it.
The best that I can think of is voltage issues or some BIOS tweaks. I may order some more memory just to rule out bad RAM.