I think the best way to test is to remove everything except the essential parts (motherboard, processor (and fan!), memory, graphics card, keyboard, monitor) and then try it. If it still doesn't work it does mean replacing each part unless someone else on here can tell you otherwise.
Having had the CPU replaced twice though it does look like the board.
Don't want to be patronising as you say it's the "first time anything like this has happened" suggesting you've done this before, but it doesn't hurt to check the obvious, e.g. if the the processor is definitely in properly, and also that the memory and everything else is properly fitted. Also there should be another 4 pin power supply connected to the board.
If it still won't work, I'd say it's more likely to be the board, so if you can't test the CPU in another machine it's probably best to take the board back for a replacement.