I clean my sink with ajax every day after washing the dishes, but that doesn't prevent this nasty build up. I have very stubborn Utah hard water.
Here is how to use cheap plain white vinegar to get rid of hard water:
For the refrigerator in-door dish:
Pop the dish out, pour a little vinegar in it and let it sit for about 20 minutes. I use an old toothbrush and the hard water just wipes off without any scrubbing.
To clean faucets:
Soak some paper towels in vinegar and placed them on the areas with hard water. Fill a spray bottle filled with vinegar and spray the towels every time you walked past. After about 20 minutes take the towels off and scrub off the hard water with an old toothbrush. For areas with thick hard water, put the towels back on, spray with more vinegar and wait another hour keeping the towels soaked with vinegar. You shouldn't have to scrub really hard, the hard water will off easily after soaking in vinegar.
Warning: Do not use vinegar on marble countertops or other stoneware, as it can cause the stone to pit and corrode.
Since writing this post, I bought a water softener and no longer need to use vinegar to remove hard water. Having a water softener has made cleaning much easier!