alaricljs's post lists pins in a new standard used by makers of some pre-built machines (HP, Gateway, etc.) for the header where the wires from front panel switches and LED's connect to the motherboard.
Check the manual for your motherboard to find the pair of pins for the front panel power switch, and be sure you have the right wire pair connected to the right pins on the header. The power function works by momentarily shorting those two pins. You can use a screwdriver or other metal object to do the same thing to test whether it's working. If the machine starts when you do this, but it doesn't start when you connect the switch to the same pins, the switch, the little two pin connector or the wire pair is bad.
Good luck. 🙂