Computer Randomly Shuts Down when upright

So this problem I'm having is really bizarre. First off, here's my system components:

EVGA X58 3x SLI motherboard
Intel Core i7 920 CPU
12GB G-Skill RAM
2 1.5TB WD Green Hard drives (RAID 1 using onboard Intel controller)
Antec Signature Series 850W PSU
Dual EVGA 9800 GT 1GB Graphics cards

Ok... that said, here's my problem. If I lay my computer down on its side, it works just fine. However, if I put it back upright, within 1 minute, it will do a hard shut down, almost as if I had held the power button down. Works every time. If I start the computer up in the upright position, sometimes it will shutdown before it even gets through the RAM check, other times it will hard shutdown after booking to Windows 7 - seems pretty random as far as the timing goes, but as soon as I lay it back down again, it works fine (I'm currently typing this with my computer in laying-on-its-side position).

Here's what I've done to troubleshoot it it - I tried pulling out the connectors to the case's power/reset switches to see if there was a problem with them, but that didn't change anything. I tried pulling out each graphics card and rebooting - still get the same problem. I ultimately pulled the whole thing apart to see if there was a loose screw or something floating around, but found nothing. I'm at a complete loss. Has anyone seen anything like this before? Any ideas or suggestions would be super helpful.

I've made a little more progress on this.

Since my last post, I downloaded EVGA's "E-Leet" tuning utility and watched the temperature monitors. When I tilt the case the the upright position, the temps start skyrocketing. I haven't let it go long enough to let it overheat to the point that it shuts down since I downloaded the utility, but because of this, I suspect overheating is the culprit. Now I'm looking for loose fan wires... I'll post again when/if I find anything.

Yep, you hit the nail on the head. It's the stock cooler and doesn't lock down properly (I've heard this has been an ongoing issue with these Core i7 fans). So... I got some more thermal compound and made sure the heat-sink was really glued down good (figure of speech, hehe) - that seems to have done the trick for now - immediately brought the idle temperature down by about 15-20C. I'm definitely going to be getting a new, better fan ASAP. Any suggestions?