We propose a new homogenization methodology for periodic electromagnetic structures (photonic crystals and metamaterials). The physical essence of this methodology is in seeking effective material parameters that provide, in some sense, the best approximation of Bloch impedances as well as dispersion relations over an ensemble of physical modes in the structure. Mathematically, the homogenization problem is solved by approximating the fields on the fine and coarse scales with the respective eigenmodes that satisfy Maxwell's equations and boundary conditions as accurately as possible.