Or you could get a bit fancier.
I've got my pacman repro hooked up to a relay powered by the pc's power supply. Since it takes a second or two to warm up, and the power supply doesn't send full voltage immediately my cab is well into the booting process before the monitor comes on. Now this probably isn't going to work for you BUT, you could take a cheap avr or maybe even a timer chip and write a simple program to turn on an output after X seconds, and leave it on after power up. Take that output, hook it up to a relay and you are golden.