In this paper we study continuous-time Markov decision processes with the average sample-path reward (ASPR) criterion and possibly unbounded transition and reward rates. We propose conditions on the system' s primitive data for the existence of-ASPR-optimal (deterministic) stationary policies in a class of randomized Markov policies satisfying some additional continuity assumptions. The proof of this fact is based on the time discretization technique, the martingale stability theory, and the concept of potential. We also provide both policy and value iteration algorithms for computing, or at least approximating, the-ASPR-optimal stationary policies. We illustrate with examples our main results as well as the dierence between the ASPR and the average expected reward criteria.