Thanks everyone for the replies.
In the process of following your advice, I realized that the CPU fan was damaged, and wasn't mounting on the board properly. As a result, the CPU was running too hot, and that's why the motherboard was shutting off. I replaced it, and now everything is fine.
I recently bought a new case (Antec Sonata II). After moving my previosly functioning PC into it, the motherboard (Asus P5B-Deluxe) started powering off randomly, usually within a minute or two of powering up.
I tried replacing the power supply with a known good one (the PS from my previous...