The problem you are having is the fill solinoid. These are located at the rear of the unit and are not too hard to change out.
Follow the line from the wall to the solinoid package. Remove the solinoids and take them to your local appliance center. They will set you up with a new one. Install it and you should be good to go.
Be sure that the filter (if so equiped) is not plugged before you remove the solinoid. The instructions for doing this should have been included with the unit when you purchased it.
Ok if it was the filter then you would not be getting drinking water. The solinoid is going to be the problem.
What you can do is raise the lever on the ice maker to immitate a full ice bin. Use something to keep it in place (some units will remain in place if pushed up till they lock) then close the door for a few minuits. Unplug the refrigerator and then move the arm back down. As you plug the refrigerator back in listen to the solinoids carefully. You should hear a click as the ice machines feed solinoid is activated. If there is no click then you will need to repeat the proceedure and use a volt meter to check for voltage to the solinoid. If no voltage is read then a service call would be in order. Any further testing is complicated and is best left to a qulified tech. I personally will not do any repairs beyond this point. If voltage is indicated then the solinoid is definatly bad and will require replacement.