There are a number of different types of ice cube tray materials which don't present these dangers.
Metal Ice Cube Trays
Stainless steel ice cube trays are safer then the plastic ice cube trays as there is no transfer of toxicity into the ice you have frozen. These metal trays also look sleeker and more upscale then your standard plastic ice cube tray. The downside to them is that they get incredibly cold once they have been in the freezer and have ice in them, obviously keeping the ice as cold as possible but also freezing your hands when you try to crack the cubes out.
Rubber Ice Cube Trays
Rubber ice cube trays are also becoming more popular, usually for party ice cubes as these trays can have shapes and designs cut into them that can make each cube look a certain way. These rubber trays are safer then plastic trays as they won't transfer the toxicity of plastic, though sometimes the ice has a hint of a rubbery taste to them. These ice cube trays are also unbreakable and will last longer then plastic ice cube trays.
Silicon Ice Cube Trays
Silicon ice cube trays have also been used instead of plastic ice cube trays, as the silicon is more durable and less toxic then plastic. However, people who have used silicon ice cube trays have complained about a white, waxy residue ending up on their glasses after washing them or small white specks in their drinks with ice cubes from the silicon trays. While it is still up for debate as to whether silicon ice cube trays are healthy, the safest thing to do would be to used metal or rubber ice cube trays.