Have a look at the power supply too, especially if the new machine has more hardware inside.
I've had more than a few PCs on campus where I work run fine, right up until you put a larger than normal load on it. As soon as you put a CD in them and it tries to spin it up, the machines shut down or reboot. Same if you plug anything into the USB ports or try to add extra drives. Replacing the power supply corrects the problem (on these at least).
-JM
