I'm sure you have already done this but unplugging and re-plugging the main MB connector can magically cure some problems . . .=)

Tried it, yeah. The only thing is that on the new PSU it's two very separate plugs that you can't join together before plugging in. So it does require some extra care when doing so. I'll check it yet again tomorrow just to be on the safe side.

Remove all the components from the motherboard, CPU, RAM, everything (except power).

Boot it up a bunch of times and note whether the motherboard complains (beeps) every time. If it doesn't, the motherboard is definitely broken and needs replacing.
If it does beep every time, put the components in one at a time until it sometimes doesn't which should indicate either the defective component or the part of the motherboard which has failed (i.e. the RAM slot itself might have a bad connection to the board).

You can see if it is the component/motherboard by testing it in another machine (providing you have one at hand that accepts it).

Thanks! I'll give this a try right after the memory testing.

Bad memory can definitely cause your computer to restart without even a single beep code or post. My friends computer would constantly restart without posting or any kind of error code. I tracked it down to one of the ram sticks causing problems with current and voltage which made the motherboard cycle on and off continuously.

Well I kind of hope it's that as it would certainly be cheaper to solve than replacing the motherboard. Easier too given that AM3 DDR2 motherboards don't seem to be in stock much at the moment.

Nah, they're pretty cheap when they're in stock.

Okay, so it seems that the RAM is completely fine. As is the previous processor actually. The problem lies in a couple of RAM slots on the motherboard. So the motherboard can't really be relied upon anymore and will need to be replaced at some point. But at least it seems that the system is stable for the time being at least.

Thanks for all your help, guys!