What would cause an icemaker to make dirty ice- The drinking water that comes out is clean and clear however when I get ice and add water to it I notice tiny little flakes and particles floating?

I just solved this problem at my house (1/08).

In some models of ice makers the Teflon coating on the tray can flake off and you will generally see small grey or black flecks in the melted ice. The solution here seems to be replacing the ice maker.

Clean the ice bin. I read of one case where the kids had placed a hotdog in the bin and it was ground up with the ice to make crushed ice.

Check if you have hard water. You may need a water softener. If you have a water softener, you may have run out of salt.

Some filter media contain fines that need to be flushed out before using the water. The fines usually appear as small black particles. If you have just changed a filter follow the manufacturer's instructions.

I have hard water, but the softener was working and had a full bin of salt. In desperation I tried freezing a small sample of water directly from the kitchen faucet and another from the under sink filtered water dispenser. When I checked the frozen samples the next day I discovered the first sample was clear as it melted in water but the ice from the filter contained small, white, pulp-like particles. I was able to solve the problem by replacing the under sink filtration system and tubing.