Your problem is most likely the icemaker itself. I had the same problem about 3 months ago, and after a lot of tinkering and wasting $55 for a repair man to come, he pointed out a simple solution.
To check, just remove the icemaker cover and jump V & L using a piece of copper wire (12 or 14 gage), and that will force the icemaker to call for water. If no water comes, there's a problem with your valve. If water comes, there's a problem with your icemaker. According to the repairman, 1 out of 200 times it's the water valve.
Solution most likely is to replace the icemaker, which will run about $100. It's very easy to remove and replace. Check the link below to see the history behind my problem.