Monte Carlo simulations of random walks in anisotropic structured media are performed to determine the dependence of effective diffusivities on geometrical properties. The anisotropic media used in this study are periodic systems, which are generated by extending primitive, face-centered, and body-centered unit cells indefinitely in all axial directions. Results of simulations compare well with published experimental data and the calculations by the volume averaging method. In addition, these results suggest that if the 2D media with percolation thresholds subtantially differ from those of 3D, 2D approximations of 3D media are not satisfactory. When percolation thresholds are the same, the effective diffusivity tensors depend solely on the porosity. This fact has been suggested for isotropic media and it seems to hold for anisotropic media.