Precipitate from Ice Cubes

A few days ago, I fixed a drink that was mostly water and ice cubes.  Yes there was vodka.  Anyhow, I noticed that a white precipitate of all sizes up to a millimeter or two was falling from the ice.  It so disgusted me that I went to the CVS store and purchased a PUR water filter for the faucet, 25% off!

After I arrived back home, I installed the filter and dumped all of ice into the sink.  Never one to pass an opportunity, I gave the food dispenser a good clean/sharpening by running the ice through it.

I fixed the same drink sans ice and all was crystal clear.  I filled my ice trays with the filtered water any the next time I made a clear drink with ice, there was that same precipitate!  After some research, I determined that it was likely calcium carbonate that I was seeing.