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This work presents a full-wave integral equation approach specifically conceived for the analysis and design of laterally-shielded rectangular dielectric waveguides, periodically loaded with planar perturbations of rectangular shape. This type of open periodic waveguide supports the propagation of leaky-wave modes, which can be used to build leaky-wave antennas which exhibit many desirable features for millimeter waveband applications. The particularities of the leaky-mode analysis theory are described in this paper, and comparisons with other methods are presented for validation purposes. Using this leaky-mode analysis method, a novel periodic leaky-wave antenna is presented and designed. This novel antenna shows some important improvements with respect to the features of previously proposed antennas. The results of the designed radiation patterns are validated with three-dimensional electromagnetic analysis using commercial software.