I believe each game has only one die. The die is a 6-sided brick with a rubber casing that protects the edges. Each side has four studs exposed, where a 2x2 brick or plate could attach. What parts are included to attach to the dice varies from game to game. For some games, it is simply six different-colored 2x2 tiles. For some games, the sides have 1x1 and 1x2 tiles of various colors. And a few come with six 2x2 white tiles with the traditional dice pips (1-6) printed on them. If you want to know which games have what I suggest BrickLink, here is a list of all sets that include the new die:
http://www.bricklink.com/catalogItemIn.asp?G=64776&colorID=5&in=A