We introduce a model for diffusion of tracer particles in cellular media in which the walls of a cell are characterized by strongly reduced permeability. Our analytical results, confirmed also by extensive Monte Carlo simulations, reveal several distinct regimes of diffusion behavior in time whereby an initially normal diffusion at very short times turns into transient one at a characteristic crossover time tau(S) and later, after a period marked by another characteristic time, tau(L), returns back to normal. At fixed permeability p of the cell. walls we find that these crossover times scale as tau(S) proportional to L(2) and tau(L) proportional to L with the size of the cells L whereas for L = constant one has tau(L) proportional to p(-1) Our results for the frequency-dependent conductivity sigma(omega) show that at low frequency the real and imaginary parts of sigma(omega) vary as omega(2) and omega, respectively, while saturating at constant values for omega --> infinity. By measurement of the dc and ac conductivity of charge carriers, it appears possible to determine both the size of the cells and the permeability of their walls.