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Eigenvalue analysis of a periodic structure by the finite-element method gives its Floquet propagation constant at a given frequency. Using this method directly to find the dispersion curve is computationally expensive, particularly in 3-D, because a large matrix eigenproblem must be solved at each frequency. The cost can be lowered by applying model-order reduction. A full-size eigenproblem at one frequency provides the information needed to build a much smaller matrix system that is sufficient for finding the dispersion over a frequency range. By controlling the frequency step-size and estimating eigenvalue errors, it is possible to compute dispersion over an arbitrary frequency range in an automatic way at a cost that is much lower than using the direct approach. Results are presented for a number of 3-D structures with rectangular cells: a triply periodic metal cube, three doubly-periodic planar structures, and a singly-periodic iris-loaded waveguide. Comparisons with previously published results demonstrate the accuracy of the method. The computational cost for these cases is at least an order of magnitude lower than the cost of solving the full eigenvalue problem at each frequency.