In this paper, we establish the first finite-time convergence result of the actor-critic algorithm for fully decentralized multi-agent reinforcement learning (MARL) problems with average reward. In this problem, a set of N agents work cooperatively to maximize the global average reward through interacting with their neighbors over a communication network. We consider a practical MARL setting, where the rewards and actions of each agent are only known to itself, and the knowledge of joint actions of the agents is not assumed. Toward this end, we propose a mini-batch Markovian sampled fully decentralized actor-critic algorithm and analyze its finite-time convergence and sample complexity. We show that the sample complexity of this algorithm is O(N^2/\epsilon^2). Interestingly, this sample complexity bound matches that of the state-of-the-art single-agent actor-critic algorithms for reinforcement learning.