Motherboard

If all the above fails then remove all ram but one stick, take all add on pci cards out, any usb you can live without, anything else that doesnt have to be on the machine to boot, even unhook the floppy, then see if it will boot.
Take ram and use a pencil eraser to clean ram where it connects to the mobo and clean vid card same way if you have one ( could be built in vid). Make sure all connections are properly connected. Also clean all dust bunnies out of the pc.
If no luck then totally disconnect the hard drive and re-connect the floppy and try using a win98 boot disk to boot to dos. Doesnt matter what os you want to install, you can use a win98 boot disk to boot to dos.
Check all pins, like monitor, ps2 mouse & keyboard, etc and make sure no pins are bent. Unhook main power from power supply to mobo and re-attach.

You could also have a bad front power\on switch. To get around that unhook
it from the mobo and use a plain flat screwdriver to touch the pins instead of the wires from the switch. If the pc boots then its the power\on switch that is bad. They are very very cheap switches and i have seen that happen!
If no luck then it might be time to take the mobo out of the case and try and boot that way.

I believe it will but you didnt say if the radeon vid card is pci or agp. If agp go to Asrock and see the agp specs for your mobo and go to ATI website and get the specs on the vid card you have. Usually they will work but i stand to be corrected, i am not a wizard on vid cards.