I'll assume that there are two players, and that you spin the cylinder at the beginning, but not after each turn. If this is the case, then at the start, the die is cast; the bullet is in one of the chambers, each with equal probability. It therefore has an equal probability of being in an even or odd chamber. Depending on if you go first or second, you either take odd-numbered or even-numbered turns. Either way, you have the same probability of losing.

On the other hand, if you spin the cylinder after each turn, it's best not to go first.