Google THE DEAD PC and you find the usual tech troubleshooting where you remove parts until it works or it's down to as few parts as you can go.
If you find removing the GPU lets it power up then that's either the GPU or the PSU at fault.
(Note. Acronyms unclear? Please google.)
If you hit rock bottom and there's only a few parts but still dead, then the bad part or parts is what was left.
I went to use the desktop and found it would not power on with the exception on a blinking light on the motherboard being accompanied by a clicking sound. I replaced the battery on the motherboard thinking this may be a signal of that, unfortunately that does not seem to have resolved the issue.
Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic