Most likely needs a ATI GPU video repair if it's not the capacitor problem mactech spoke of. Open the machine by taking out the three screws on the bottom of the frame and examine the cans for corrosion of the tops, or bulging. Please take a look at this post for comments on problems on this type machine.

http://www.ifixit.com/Answers/View/4752/...

You can try DT&T services to get a replacement board installed at DT&T service.

http://www.dttservice.com/iMac.html

But I have a unit that came in from a client that had the board replaced by Apple and it failed after a year. The G5 iMac is really a waist of money. Would suggest cut your loss and buy a new intel iMac. The unit is strong and reliable.

There is a kit you can get to replace the bad Capacitors on the logic board which is most likely the problem but it is hit and miss. Here are the instructions:

http://www.jimwarholic.com/labels/iMac_G...

ken