A new method is presented for a systematic calculation of the effective dielectric constant e of a granular, two-component composite material with a precisely known microscopic geometry. The basic approach is to attempt to calculate the poles and residues of e as a function of the dielectric constants of the pure components. This calculation is reduced to an eigenvalue problem with a non-negative, self-adjoint, bounded linear operator. That problem is dealt with in two stages: First the eigenvalue problem for each of the individual, isolated grains is solved. Then the general eigenvalue problem for the entire composite is expanded in terms of the individual grain eigenfunctions, and in this way it becomes a matrix elgenvalue problem. The method is applied to a number of physical systems, including a general periodic composite, and a simple-cubic lattice of identical spheres. The results include some new predictions concerning additional optical resonances which should appear in periodic or quasiperiodic metal-insulator granular composites.