I discovered that the principle cause of Mac IIcx failures is leaking capacitors on the motherboard. Open the machine up, unplug and remove the drives & power supply and have a look on the motherboard for electrolyte residue (it will look a bit like spilled cola). This can easy be cleaned off using some Q-tips and cleaning alcohol such as isopropanol.

A new PRAM battery wouldn't hurt either, but it's not essential for booting.

My IIcx had the appearance of being completely dead until I cleaned the motherboard. It now runs perfectly!