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A new numerical method is presented which is suitable for the analysis of scattering by dielectric structures which are periodic in one dimension. This method is based on a rigorous differential equation formulation of the scattering problem. It is particularly appropriate for structures which have periods on the order of the incident wavelength, and which have a depth of several wavelengths or less. Results are presented for periodic dielectric structures for TE polarization. Comparisons with results that have been obtained by other authors, and with a case involving inhomogeneous dielectric for which the exact solution can be found, are made for transmission type dielectric structures. These results establish the accuracy of the method. Further results are presented for reflection structures consisting of arrays of rectangular dielectric cylinders over a ground plane, and for transmission structures consisting of arrays of rectangular cylinders in free space, as well as for transmission structures consisting of semi-infinite dielectric media with sinusoidal or rectangular surface height variation.