For all of you having this problem:
I recently got an old PC with the exact same problem and I fixed it. I don't know if it's the solution for all the cases, but it's a cheap option before trashing your motherboard or PSU.
You need a multimeter, a soldering iron, a lamp and some capacitors.
1. First of all, you should check your PSU. Take a paper-clip, straighten it and short the green and a black wire of the 20-24 ATX connector. The PSU fan should start working. Don't use the multimeter yet, many faulty PSU's work normally under zero load.
2. Take a 12v lamp (you could use one of your car's brake lights). Solder two pieces of wire on the lamp and connect the other ends on your multimeter's leads.
3. Set your multimeter on 20V and connect the leads on:
a. Red lead on yellow wire, black lead on black wire. The lamp should light up and your multimeter should read between 11,2 and 11,8 volts.
b. Red lead on red wire, black lead on black wire. The lamp will barely light up and your multimeter should read between 4,2 and 4.8 volts.
c. Red lead on orange wire, black lead on black wire. The lamp will not light up and your multimeter should read between 2,7 and 3,1 volts.
d. Red lead on black wire, black lead on blue wire. The lamp will light up and your multimeter should read between 11 and 11.5 volts.
e. Red lead on black wire, black lead on white wire. The lamp will barely light up and your multimeter should read between 4,2 and 4,8 volts.
Don't test the grey and purple wires! You could damage your PSU. Also, don't use the black wire that is short with the green one. If the voltage ratings are somewhere between those given, then there is nothing wrong with your PSU.
4. Look carefully on your motherboard for something like this:
If any of the capacitors has a bulging top, a yellow or white stain or a broken shell, then replace it with a new one. This step requires some experience, if you are not sure what you are doing, let someone else do it for you. The problem in 95% of non-booting motherboards is leaking capacitors.
5. If you replaced the capacitors and the motherboard won't boot, then throw it away, the problem is some IC or the flash BIOS memory.