Derbykidd - You've left out the video card. Here is a link that will give you a rough idea of your power needs by plugging in your system specs.
http://www.jscustompcs.com/power_supply/Power_Supply_Calculator.php?cmd=INTEL
I plugged a 9800XT in there just because it would seem to be in-line with your specs and the maximum power needed is 345 watts. That is with everything turning and burning simultaneously, so it very well may be your PSU if the installed unit is of poor quality and isn't really producing 350 watts continuously.
Try the below and se if you can get your system to boot:
You are going to have to do a barebones setup and test each component. This will read a lot harder than it actually is. The initial procedure takes only around 10-15 minutes. The follow on troubleshooting may take a lot longer though. Also, please do not skip steps. Do everything in order and as listed or your troubleshooting will be flawed.
Caution: Please remember that turning a PC off does not mean there is no power going through it. Modern systems maintain a trickle of power to keep the standby functions running. You either have to turn off the switch on the Power Supply Unit (PSU) itself or unplug the system from the wall. Unplugging is best. If you have a LED on the mobo that is lighted all the time. make sure it is out before proceeding. Also, be aware of static. Make sure you wear and ESD strap or discharge yourself on a steel part of the case before touching anything inside.
First, unplug the PC from the wall and then open it up. Disconnect all the drives (floppy, CDROM, DVD etc.)from the motherboard (mobo) and also disconnect your Hard Drive(s) from the mobo. Do not leave the hard drives connected. The system will boot into BIOS just fine with no hard drive attached. Unplug the power from all those drives you disconnected from the mobo. Remember to disconnect the front panel firewire and/or USB ports.
Next, remove all the RAM, except for one stick, from the mobo. Some mobos are very picky about where the RAM needs to be placed so make sure the one stick of RAM is in the correct slot as per your manual.
Now you are stripped down to a barebones system. The PSU, the mobo itself, 1 stick of RAM, the CPU/HSF and video card. Reset your CMOS/BIOS while the system is stripped down, unplugged and open. You do this by removing the battery and then moving a jumper near the battery around. Usually there are a set of three pins with two covered by a jumper. You move the jumper from pins 1&2 to pins 2&3 and let it set for a few minutes then reset the jumper to pins 1&2 and replace the battery. CMOS and BIOS will be back at default settings after doing this.
Now check that everything is seated correctly, both the 4 pin and 20 or 24 pin power is connected and secure and if so then plug the PC back into the wall and make sure that any LEDs that should be lighted on the mobo are lighted. If all is still well then turn it on. Hopefully she boots right back into BIOS.
If you get back into BIOS you can start troubleshooting by turning the PC off and unplugging it and reconnecting peripherals one at a time. The idea here is to connect and reboot until something hangs your system up This presumably is the bad piece of gear.
If you cant get into BIOS and have the same problem as before then you know it is either the PSU, the RAM, the CPU, the mobo itself or the video card. Change out each these until you get into BIOS. I would start at the PSU as it is usually the guilty party in a situation like this and is also easy to change in and out you are down to just 2 plugs now remember). Next up would be the video card and/or RAM and if still no luck then things get hard as you now have to consider either the CPU or the mobo.
Good luck and happy hunting.
