I replaced the motherboard, and that didn't fix the problem, but I needed a new mobo or a new sound card anyhow. This was a big upgrade to my old mobo, so I just kept it.

I replaced the power supply. That was causing part of the problem, as the computer stayed on longer afterwards, but it was still shutting down.

Guessed that my heat sensor wasn't updating fast enough and that my processor was the reason my computer was shutting down. I replaced the heat sink and fan, and it's fine now.

Also, during Windows installation (on a CD that I've used a couple of times before with no problem) it froze twice around 60%. A new CD fixed that problem.

What is it they say about silence being golden?