Another of those infamous random freezes killed my computer this morning. This was after I'd spent all night running Memtest86+ to make sure I wasn't dealing with a rogue bad-memory issue; I got through eight passes with the program (with cache both on and off) and found no problems.
Whenever I've dealt with a problem like this in the past, my usual way of debugging it has been to first remove everything that does not absolutely have to be there, and then cycle out the components that can be changed. Since I no longer have any expansion cards left in the system, I decided to cycle out the video card (an ATI Radeon 9550) and replace it with an older one that had originally shipped with the PC (an NVIDIA Quadro FX 500). That meant losing Aero for the time being, but I'm patient and losing Aero for a bit is not going to break me.
I'm now tentatively wondering if the Radeon a) had some kind of "sleeper" issue that was only exposed when Vista started really stressing it a bit, or b) it was already starting to malfunction and it did so coincidentally with my move to Vista. It's a bit of a parallel with the folks who moved to Vista and then discovered one of the DIMMs in the PC in question was bad -- and that this only surfaced with Vista because of the ASLR (address space layout randomization) system in use. Likewise, maybe there was a defect with the card that only surfaced in the rarest circumstances before, but is now surfacing more regularly with Vista due to differences in the way the card's being handled. It's tempting to blame Vista, but off-base; it doesn't solve anything.
If I go for three days without a crash using this older card, I'll swap the other one back in and try it without Aero to see if that makes any difference. Either way, I think I may be looking at a new video card.
One last possibility is that I might be looking at some weird side effect of DEP. If I continue having problems, I might try turning DEP off (or put the old card back in and try it with DEP off to see if that affects anything). A shot in the dark, but a shot worth taking.

Serdar,
It's also possible that you have a power supply issue, so you might investigate that if your other investigation doesn't reveal the culprit.
You might also try booting with a Live CD and running something processor intensive to remove the OS from the equation.
[Thanks for the suggestion! The MemTest86+ test was a standalone DOS application, but a live CD doesn't sound like a bad idea. As I write this, however, I've been running for almost two straight days with no problems after having swapped out the video card. The power supply's rated for 600 watts, though, and since the original system was AlienWare I'm inclined to think the problem lies with the video card (which was my own add-on) and not the juice. -- ed]