iPod Touch 'apple' screen cycles on and off, restore/firmware update doesn't fix

iPod touch 3g 'apple' screen (NOT the white screen of death) cycles on and off the display. Have tried factory restore and firmware update in iTunes several times on two different computers. The update proceeds, but iTunes doesn't recognise the device afterwards, and apple screen keeps cycling. The innards are all connected, no cables torn, etc. Any ideas for repair options? Worth trying to source a replacement logic board?