In this work we suggest a model for diffusion of particles in cellular media in which the walls of cells are characterized by strongly reduced permeability. Our analytical results are obtained for a regular system and confirmed also by extensive Monte Carlo simulations. They reveal several distinct regimes of diffusion behavior in time whereby an initially normal diffusion at very short times turns into a transient one at a characteristic crossover time S and later, after a period marked by another characteristic time L, returns to normal. At fixed permeability p of the cell walls we find that these crossover times scale as SL2 andLL with the cell size L, whereas for L=const one has Lp-1. These transitions from Gaussian to transient behavior are analyzed by cumulants of the mean quartic displacement x4(t). Our results are valid for a regular arrangement of walls; however, we find generally that the course of the mean-square displacement x2(t) with time is very similar to results obtained in the past for diffusion in disordered media. The frequency-dependent conductivity () shows that at low frequency the real and imaginary parts of () vary as 2 and, respectively, while saturating at constant values for . By measuring the dc and ac conductivity of charge carriers it becomes possible to determine both the size of the cells and the permeability of their walls.