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In a previous work, a conceptually simple method for incorporating periodic boundary conditions into the finite-difference time-domain method was introduced. That work was limited to two-dimensional problems that were periodic in a single dimension. In this work, the method is extended to the more general case of three-dimensional problems that are periodic in two dimensions. For scattering problems, the computational cost of the method is shown to depend on the direction of the incident plane wave. The method is demonstrated by calculating the reflection coefficient from a dielectric slab and by modeling two perfectly conducting, infinitesimally thin periodic structures that are complementary to each other. For the complementary structures, the results computed by this method are shown to be related through Babinet's principle.