In this paper, we show how lattice–based random walks of virtual particles directed by Monte Carlo methods (Lattice Monte Carlo) can be used to address a variety of complex phenomenologically mass diffusion problems. Emphasis is put on the practical details of doing the calculations. It is shown how concentration depth profiles can be determined: this is exemplified with diffusion in the presence of isolated dislocation pipes, grain boundary slabs, and oxygen segregation at interfaces in metal-ceramic oxide composites. It is also shown how effective diffusivities can be determined in materials. We also show how temperature profiles and the effective thermal conductivity can be determined for the thermal diffusion analogue of mass diffusion. A detailed comparison is made in one case with the results of the Finite Element method.