In this work we establish the posterior consistency for a parametrized family of partially observed, fully dominated Markov models. As a main assumption, we suppose that the prior distribution assigns positive probability to all neighborhoods of the true parameter, for a distance induced by the expected Kullback-Leibler divergence between the family members' Markov transition densities. This assumption is easily checked in general. In addition, under some additional, mild assumptions we show that the posterior consistency is implied by the consistency of the maximum likelihood estimator. The latter has recently been established also for models with non-compact state space. The result is then extended to possibly non-compact parameter spaces and non-stationary observations. Finally, we check our assumptions on examples including the partially observed Gaussian linear model with correlated noise and a widely used stochastic volatility model.