I ordered a new ice maker and installed it. It did not work either. So I ordered a schematic from these guys and was able to see why it did not work. The bimetal thermostatic switch was not operational at the temperature of the freezer....-5deg. Something is not right with it! So I used the switch from the broken icemaker and put it in the replacement one and we now make ice.
My hunch is that the old one that made the clicking sound was that of the motor and gearing just worn out. The new one was OK except for the bimetal switch.
I was able to diagnose this problem by someone who was kind enough to give several diagnostic tests on the net by using a meter to find the problem. Wheather it is a bad motor, bad tray warmer, or bad thermostatic switch.
The unit is designed to dispense water and freeze it in the ice trays. While this is taking place the warm water warms the ice tray and opens the bimetal switch which stops the clock motor from running. Once the temperature of the ice tray reaches 10-15 degrees "frozen Ice" the bimetal switch then closes and turns on the clock motor to warm the ice tray slightly, and then turns the motor on which extracts the ice and allows the cycle to start all over again.
This only took me 4 days to figure out! But now it works and now, I really know why!