You had a collection of parts that didn't work right. You changed one of the parts and it still doesn't work right, so assume that the part you changed wasn't the bad part to begin with.
The computer would never booted into the BIOS if the PS or CPU had hard faults. RAM is usually the hardware culprit for unstable systems. Try using one stick of RAM in each socket until the PC boots into the BIOS. If the first stick won't boot in any socket try the other. (Keep the HD and optical drive disconnected while doing this, the fewer the number of variables, the better. )
If there's still no joy, try connection the PS to a working system. This won't take long because you don't even have to put it in the case.
That should keep you busy for a while, so came back when you're done.
The parts were mostly working pulls from another system I have that I had been upgrading incrementally over time. The RAM I actually removed from my desktop when I assumed this system. I took the opportunity to go from 4x1GB to 2x2GB, with an upgrade to 8GB planned in the future.
Anyway, I will assume that the previous motherboard could have killed one of the sticks and try experimenting with them. If that does not get it to start (without turning off within 1 second of starting), I will try a new PSU. I do not want to put the PSU in my desktop if there is a chance it might damage it. Does anyone have any suggestions on where I can get a low cost PSU that would definitely not run the risk of possibly tripping current underflow protection in my system? If the RAM turns out not to be the case, I would like the PSU I buy to be a Seasonic PSU.