For continuing tasks, average cost Markov decision processes have well- documented value and can be solved using efficient algorithms. However, it explicitly assumes that the agent is risk-neutral. In this work, we extend risk-neutral algorithms to accommodate the more general class of dynamic risk measures. Specifically, we propose a relative value iteration (RVI) algorithm for planning and design two model-free Q-learning algorithms, namely a generic algorithm based on the multi-level Monte Carlo (MLMC) method, and an off-policy algorithm dedicated to utility-base shortfall risk measures. Both the RVI and MLMC-based Q-learning algorithms are proven to converge to optimality. Numerical experiments validate our analysis, confirm empirically the convergence of the off-policy algorithm, and demonstrate that our approach enables the identification of policies that are finely tuned to the intricate risk-awareness of the agent that they serve.