You need to get a Power Supply Tester. It is a little "box" that has power connectors for Floppy, SATA, PATA, and various MB connectors. You plug your Power Supply cables into it, and power up your PSU and the tester will let you know if everything is OK.
Since you said the PSU still works with the old parts, then I assume the tester will show everything is OK with the PSU.
Then, with the PSU power off, while the tester is still attached to the PSU, remove the MB power connector from the tester and attach it to your new MB and recheck the power.
- The -12V Power is only present on the MB connector, so once you remove the MB connector from the tester and plug it into the MB, the tester will say the -12V has failed.
- Also the +3.3V is only on the MB and SATA cables. So if your PSU doesn't have a SATA cable to plug into the tester, when you remove the MB connector from the tester and plug it into the MB, the tester will say the +3.3V has also failed.
Ignore these for the moment as long as all the other voltages are still showing as good.
If everything is OK when the MB is under power, power off the PSU and remove the power cable from the MB and re-attach the power cable to the tester.
Then, one by one, attach an unused power connector to one of the parts (drives) from your new setup and recheck the power. If everything is OK when that part is under power, power off the PSU and remove the power cable from that part and try the next part.
At this point, only check one new part at a time. After each part has passed individually, you can start adding power to all the parts, adding one at a time, while still keeping as many connectors attached to the tester as possible (add MB last).
If (when) the PSU fails to start, the tester should tell you which power rail has failed (+3.3V, +5V, +12V, -12V, -5V).
By now, you may have identified a part that makes the system fail to start.
My guess is, that while your PSU looks to be working, and can supply enough power for each of the output voltages to run your old parts, the new parts probably use more of one of the output voltages (perhaps barely more) than your PSU can handle or more than it was designed for.
Just for example, suppose your PSU can supply +3.3V at 1.0 Amp. That would have been enough for your old parts. But your new parts require +3.3V at 2.5 Amps. That's a difference of only about 5 watts, but it could fail because of that. Now, this may be all that your PSU was designed to provide for the +3.3V so the PSU may be working fine but not work with all of your new parts together.
A PSU tester is handy to have around and I would recommend you get one. The one I have cost about $15 at Tiger Direct (also Fry's). You could get one with more advanced features (Digital display, etc) for obviously more money.
The next step would be to try a new PSU on the new parts. You could buy a new PSU and try it, then return it if it didn't work.