If you're PC is at least starting to post, you can rule out drives, ram, and most likely video. Basically you're down to motherboard, cpu, or power supply. So, as someone else noted, give specs of at least those 3 components. It would also be good to know what all your power supply is running - fans, usb ports (what peripherals), pci cards, etc.
At this point, with the current issue you're having (define further how far it gets in the POST process), I'd guess one of the following is happening
1) You've replaced the mobo which obviously required reseating everything. It's possible you're CPU fan/heatsink aren't properly seated and/or the heatsink doesn't have a good bond to the cpu. You could be overheating the cpu (which doesn't take long) and thats why the PC is powering off/rebooting. Check the cpu placement and heatsink. Are you using something like arctic silver to bond the heatsink and cpu?
2) If the CPU has overheated, depending on how many times or for an extended period of time, it could be permanently damaged at this point. Do you have another CPU you could test with?
3) The previous motherboard issues you had could have damaged another part - like the CPU. Again, do you have another CPU to swap out?
4) As SOSTrooper noted, your PSU may be faulty and the motherboard could be fried again. And that could have been the cause of your original issue resulting in the motherboard replacement. One of the 12v rails may be dead. The PSU may not be correctly supplying power resulting in over or under voltage for the components. Again, need to know the details from the power supply. Previously you said you had another power supply you swapped out? What are the specs of that power supply? Not only the overall wattage, but specs from the 3v, 5v, and 12v outputs.
Sorry to hear you're having issues but with some more details we should be able to help you out. Good luck!