Random computer reboots

[dotHectate]

I've got a computer that I've been attempting to resurrect from it's poor condition that I received it (free) in. It was horribly dusty, to the point where the PSU died shortly after I turned it on for the first time.
I cleaned it out real good-like and replaced the PSU (thanks RadDad). After finally re-wiring everything I was going to install XP on it, but I can't.
Whenever I turn it on, it randomly reboots. I don't know if there's a short somewhere or what, but it will run anywhere from between 1 second (or less really) to 3 minutes before just rebooting itself. There doesn't seem to be anything that is triggering it that I can see. I disconnected the reset switch to make sure that wasn't the cause. I thought maybe the CDRom was doing it because by the time it got spinning up real good it would reboot, but then sometimes it doesn't, because it's managed to get most of the way through the early stages of XP's setup.
Is there anything that might be causing this? I can't seem to find anything in the BIOS that might help (there's a "Halt on errors" setting that fiddling with didn't affect anything).
The computer is an unknown mobo, Athlon XP 2000+, with an indeterminable amount of memory (two sticks, of varying sizes. I wrote down what the mem check came up to but I didn't bring it to work with me), random CDrom, floppy, and HDD. Sorry that this isn't very detailed, but like I said, it was free crap (literally) that apparently is mostly cheap grade.
Any thoughts?

Forgive me for posting before thinking
Google search leads me to believe that the four causes are PSU (it's new), RAM (it's not, need to check this part), drivers (boot from CD? not likely that this is it), and viruses (ditto).

Any other thoughts are appreciated too. Sorry.

 

[dotHectate]

Not sure if it's the RAM or not. It was doing the rebooting thing even without any RAM in there, or either one installed, or both. It's not the CDrom, HDD, or floppy of course, because they don't have any affect when they're connected or not.
So either the MOBO is bad, the CPU is bad, the PSU is bad, or some connection between them is bad.

The strange thing is that when I first turn it on (after being off for a while) you don't even get to the "boot beep" on the pc speaker. It just reboots it self over and over again. Little by little it starts running a little bit longer each time. Eventually it will actually boot all the way into an OS or the XP install CD program, but never any farther then that. Even weirder is the fact that it can sit in the BIOS and not do anything.
Is my CPU fried?