It is known that an equilibrium of an infinitely repeated two-player game (with limit average payoffs) can be computed in polynomial time, as follows: according to the folk theorem, we compute minimax strategies for both players to calculate the punishment values, and subsequently find a mixture over outcomes that exceeds these punishment values. However, for very large games, even computing minimax strategies can be prohibitive. In this paper, we propose an algorithmic framework for computing equilibria of repeated games that does not require linear programming and that does not necessarily need to inspect all payoffs of the game. This algorithm necessarily sometimes fails to compute an equilibrium, but we mathematically demonstrate that most of the time it succeeds quickly on uniformly random games, and experimentally demonstrate this for other classes of games. This also holds for games with more than two players, for which no efficient general algorithms are known.