Strip all unnecessary hardware out of the computer - soundcards, network cards, hard drives - everything. Leave just the motherboard, processor, memory and video card. If it still doesn't power up remove the video card and listen for the "No video card" POST beeps. If it still doesn't work try removing the memory and listening for the "No memory" POST beeps.
If stripping it down to the bare bones makes it work then start addnig the extra hardware back one item at a time until it fails - that last pece of hardware is the culpret.
If nothing makes it show any life then it's either the motherboard, CPU or power supply. Try replacing them one at a time until you get something.
If you're feeling adventurous you can test the PSU by unplugging it from everything (even the motherboard) and connecting it to one spare device (an old hard drive would be ideal) to give it some load. You can then insert a paperclip into the large motherboard connector linking a green wire (pin 14) to a black wire (pin 15) - this should force the power supply to turn on and power up the device you have connected.