A dynamic lattice Monte Carlo (DLMC) simulation approach to the description of ion transport in dielectric environments is presented. Conventional approaches using periodic boundary conditions are inefficient for nonequilibrium situations in inhomogeneous systems. Instead, the simulated system is embedded in a bigger system that determines the average electrostatic potential and the ionic concentrations at its boundaries. Two issues are of special importance: implementing the given boundary conditions in the treatment of dynamical processes at and near the boundaries, and efficient evaluation of ion-ion interaction in the heterogeneous dielectric medium during the Monte Carlo simulation. The performance of the method is checked by comparing numerical results to exact solutions for simple geometries, and to mean field (Poisson-Nernst-Planck, PNP) theory in a system where the latter should provide a reasonable description. Other examples in which the PNP theory fails in various degrees are shown and discussed. In particular, PNP results deviate considerably from the DLMC dynamics for ion transport through rigid narrow membrane channels with large disparity between the dielectric constants of the protein and the water environments.