Recycling that frequently probably means you are losing the "power good" signal during startup. The power supply was a real good guess, but here's a tip that may save money in the future:

Power down, unplug one device (starting with your CD or DVD-ROM) and try a startup.
Do this with each of your devices, one at a time, finally ending with any cards you have plugged in, using the same method there. If you have a video card and it is not on your motherboard, try a spare if you have one- otherwise listen and see if there is any difference. If it is device related, you should be able to nail it.

One other thing that can (rarely) cause power recycling is a bad memory module- try removing one at a time if you have more than one. If you only have one, remove it- you will get a memory error, but see if the PC still recycles.

A jammed reset button may also cause this, or also a bad or jammed power button on an ATX system. If your power supply connects to the motherboard with a large rectangular connector, it is ATX.

After you have eliminated any hardware causing the problem, the last thing I can think of trying is resetting to BIOS defaults. Power down, unplug the AC power connector, and locate the RESET CMOS jumper, usually near the battery. Move it from (on most motherboards) pin 1-2 and put it back in across pin 2-3 for about 30 seconds, then move it back to pins 1-2 (some motherboards use the opposite arrangement- just reverse the pin order if that is the case on your board).

If none of this helps, the best advice I can offer is to browse e-bay for a BIOS test card that plugs into your PC. I did a "buy it now" and got 2 of them for $1.99 each.... this card will display a code that a cheat sheet that comes with the card will tell you where the problem is on your motherboard, which should tell you if it is something fixable.

Good Luck!