In this work, we develop a systematic and self-consistent approach to homogenize arbitrary nonmagnetic periodic metamaterials. The proposed method does not rely on the solution of an eigenvalue problem and can fully characterize the effects of frequency dispersion, magnetoelectric coupling, and spatial dispersion, even in frequency band gaps or when the materials are lossy. We formulate a homogenization problem to characterize a generic microstructured artificial material, and demonstrate that it is equivalent to an integral-differential system. We prove that this complex system can be reduced to a standard integral equation and solved using standard methods. To illustrate the application of the proposed method, we homogenize several important metamaterial configurations involving split-ring resonators and metallic wires.