ANSWER:
I have a feeling that you need to replace the filter that sits between the water source and the ice-maker itself. That filter needs to be changed routinely, in the same way you change the filter on your furnace and air conditioner.
Look in the refrigerator's owners manual for specific instructions on how to do this. You should be able to pull the appliance out from the wall, follow that waterline back to where it's hooked up (probably under the sink) and find the filter in the process.