Computer acting up

My home computer is acting up. It will boot and run for a while, then just power off. It won't respond to the power button (immediately) after this happens, but if left for a while, will power on and run for a while again. It really feels like something (maybe the power supply itself) is overheating. I'm going to check for clogged fans and stuff, and if that doesn't work, probably look at replacing the whole thing. Joy.