I took apart my existing ice maker and noticed that the turning rod had pushed through the front of the plastic on the ice maker. I believe this shift, by about 1/2" caused the metal pieces that engage the rear of the ice maker, the plastic housing and thus the rod to skip, or not get leverage to turn.
I at first tried to replace the rear plastic "cup" and clips on the front to hold the rod back, but it didn't work. I finally just broke down and ordered an entirely new assembly, and now things work great again.