Chewing ice cubes is bad for humans because it can cause microfractures in your teeth, which can eventually lead to bigger fractures. I would assume this is the same for dogs although I understand their love for them. My dog thinks they're toys and just plays with them.
In order to really enjoy a dog, one doesn't merely try to train him to be semi human. The point of it is to open oneself to the possibility of becoming partly a dog. ~Edward Hoagland
If I'm not mistaken, ice cube eating has been linked to causing bloat.. as well as ice water. It's more common in certain breeds, but all breeds -can- suffer from it. If not noticed, and treated it's absolutely fatal. Not to mention a choking hazard.