A general theory of electroosmotic phenomena in model porous media possessing spatially periodic structure is developed. General expressions for the electric conductivity, permeability, and coupling electroosmotic tensor coefficients are obtained in terms of solutions of several transport unit cell problems, posed for the linearized electrokinetic equations. By means of a consistent application of the homogenization methods, the Darcy-scale equations describing the electroosmotic flow in a spatially periodic porous medium are obtained. A numerical code has been built, with an overall precision better than a few percent, when the double-layer thickness is larger than the elementary grid size. Several model porous media have been systematically investigated, including cubic and orthorhombic arrays of spherical and ellipsoidal particles, random packing of such particles, and reconstructed media. A new correlation between the electroosmotic coupling coefficient and permeability is proposed for thick double layers.