I have had similar experiences with my old computer-- but instead of suspending or blacking out, it just completely froze at random times. What I found was rhat my problem was due in part to heat, because myo room is really hot and it crashed quite a bit less when I stuck a big blowing into the case.
You can't always just physically feel the components to tell if they are overheating. In some cases, checking the BIOS right after the computer crashes will reveal an obvious problem, but that is not always the case. I assume from your description of your solutions that you are currently running with the case open. In that case, the first thing to try would be to either stick a fan in front of your computer, or if the room is excessively hot, just open a window or move the computer to another room.
I also suspect, however, that there could be a problem or defectin your mobo or CPU. You can obviously test this easily if you have another mobo or cpu you can substiture in for the current. Otherwise, it could be a real pain to find out. Also try removing one of the processors, see if that fixes the problem (I hope I didn't misread that-- this is a dual processor system, correct?). Along the same lines, you should try swapping out every single peice of hardware you have, if at all possible, and just rule out everything you can.
Besides this, the best thing to do is just to try random things (take the whole thing apart and put it back together, try using a different OS like knoppix, a live linux distribution, meaning you can run it off a bootable cd without installing, anything else you can think of). This is the fun part of working with computers, the tedious troubleshooting of some strange problem.
And no, I'm not sure I'm right, in fact, it wouldn't surprise me in the least bit of someone else knew a much easier solution right away after looking at the problem (I personally don't have a ton of experience with the specific hardware you are using, except, of course, the P3s).
