This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the' drift and monotonicity' conditions for continuoustime Markov processes, we give conditions on the controlled system' s primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a' martingale characterization' of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.