Water diffusion in a tissue model is studied both analytically and numerically. Tissue is regarded as a periodic array of boxes surrounded by partially permeable membranes (cells), embedded in an extracellular medium. intracellular and extracellular diffusion coefficients may differ. Expressions for the apparent diffusion coefficients (ADC) in isotropic and nonisotropic tissues are derived and compared with Monte Carlo simulations. Calculated ADCs disagree with values obtained from the widely used "fast exchange" formula. Effects of differences between intracellular and extracellular T2 relaxation times on measured values of ADC and T2 are discussed. The general analysis is specifically applied to the changes occurring in ADC following ischemic insults to brain tissue. It is found that although membranes affect ADC significantly, the observed changes in diffusion cannot be due to reduced membrane permeabilities. They may result from the combined effect of changes in cellular volume fraction, extracellular and intracellular diffusion.