The performance of the generalized belief propagation algorithm to compute the noiseless capacity and mutual information rates of finite-size two-dimensional and three-dimensional run-length limited constraints is investigated. In both cases, the problem is reduced to estimating the partition function of graphical models with cycles. The partition function is then estimated using the region-based free energy approximation technique. For each constraint, a method is proposed to choose the basic regions and to construct the region graph which provides the graphical framework to run the generalized belief propagation algorithm. Simulation results for the noiseless capacity of different constraints as a function of the size of the channel are reported. In the cases that tight lower and upper bounds on the Shannon capacity exist, convergence to the Shannon capacity is discussed. For noisy constrained channels, simulation results are reported for mutual information rates as a function of signal-to-noise ratio.