Exploring the posterior distribution in Bayesian inverse problems can quickly exceed available computationally resources if each forward-model solve is computationally demanding. In many situations, however, there is not only the expensive full-forward model available. Rather, there exists a hierarchy of models that describe the same phenomenon as the full model but with varying computational costs and accuracies. For instance, approximations obtained by the grid coarsening. We present some recent advances on accelerating Markov chain Monte Carlo based samplers for solving Bayesian inverse problems by exploiting the structure of available model hierarchies.