I just built a computer from spare parts I had lying around, I can get specs if anyone really wants them.

The computer sporadically shuts itself down after its been on for a while(max it has been running for about 3 hours), and when I go to restart it right after, it only runs for about 5 min, and each time immediately following the initial shutdown I restart it, it stays on for less amount of time. To me this sounds like a overheating issue, but I have 3 120mm fans in the case, and a 80mm on the video card, with a normal heat-sink on a AMD processor. It is not overclocked and I am not a heavy gamer. From what I feel it is not overheating, at least from what I can tell.
The computer doesn't crash and go to a black screen, it literally just powers itself off, instantly no matter what I was doing.

The current OS is linux mint 13 maya edition, and i know this isn't a mint forum, but since mint is based off of ubuntu. I was hoping someone could tell me where to start looking for error logs that might tell me why my computer is shutting down randomly. Feel free to ask me for more information if needed. I appreciate all input