I will describe the analytical basis behind a fast method of computing periodic Green functions, ultimately for the purpose of efficiently solving problems of scattering by periodic structures. The Poisson summation formula provides super-algebraic convergence away from frequencies for which one of the Rayleigh-Bloch modes is grazing. At grazing (cutoff) frequencies, the periodic Green function ceases to exist, and a more complicated method is needed. This involves introducing several sheets of periodic sources to create a half-space Green function. This is work with Oscar Bruno, Catalin Turc, and Stephanos Venakides.