The periodic viaduct in this study was defined as a new kind of periodic structure-the "open"-type periodic structure-for the first time. Knowledge about the energy bands of this kind of "open" periodic viaduct is important for its aseismic design. Using the transfer matrix method and the compliances for the pile foundations, the impedances for the piers were obtained. Based on the Bloch theorem and the transfer matrix method, the nonlinear polynomial eigenvalue equation for the energy bands of the periodic viaduct undergoing in-plane motion was derived using the impedance of the piers. Based on the obtained nonlinear eigenvalue equation, the approximate linear eigenvalue equation for the periodic viaduct was obtained, and numerical results for the energy bands of the periodic viaduct were presented. The numerical results in this paper demonstrate that when the periodic viaduct is undergoing in-plane motion, there exist three lattice waves: the first kind of wave is a highly decaying wave; the second kind of lattice wave can propagate only at some frequency ranges; and the third kind of lattice wave can propagate at most frequencies.