Are we talking "turned off" as in powered off or "no signal" on the monitor?
If indeed the computer will not power back on, the list of things to check is:
1. If computer is plugged into a surge protector, ensure that the trip/reset button (if available)isn't raised.
2. If computer is plugged into a GFI wall outlet, check to see if the outlet tripped.
3. Try replacing the power cable (the one from the wall/outlet to the back of the PSU) with the power cable for the monitor.
4. Lastly, I would test the PSU. The paper clip trick could verify if the PSU can power on. Disconnect the P1 (24-pin) connector from the mobo. Get a paper clip and jump the green wire with the black wire right next to it.
If this is a "no signal"/"no display" issue, then try this list:
1. Clear CMOS by removing the battery on the mobo for about a minute.
2. Connect the monitor to the integrated VGA (if available)
3. Try a different GPU.