Re: computer turns on and off and on again
Doesn't matter if it turns on and the fan is spinning; if it's defective and not putting out enough power because of that, your PC isn't going to turn on properly. Hell, no PC it's hooked up to will turn on properly. Bad components are bad components no matter how insignificant of a defect they have. It only takes one bad capacitor to completely cripple an electrical device.
Since you do have a second computer; swap your new PSU into it and see if it will power on properly. As long as it doesn't require more power than your new computer (which I seriously doubt it would..), it should power on the other computer. If it works in the second computer but not in the first, your new motherboard is the problem.