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The phenomena of high frequency-selective transmission of a plane wave by a dielectric two-dimensional (2-D) periodic waveguide, comprising a uniform dielectric layer sandwiched by two finite thickness 2-D periodic structures served as the waveguide wall is described. This structure is termed a leaky parallel-plate-like waveguide because the waveguide walls are not perfect reflection mirrors. The scattering characteristics and dispersion relation, including the phase and attenuation constants, of the 2-D periodic waveguide are thoroughly analyzed with the modal transmission-line method and Floquet theory. The extraordinary open stopbands caused by the contra-flow coupling between a leaky parallel-plate-like waveguide and the leaky waves, which are generated by 2-D periodic structures (waveguide walls), are displayed in the form of the Brillouin diagram. The phase-match condition is used to verify the resonant coupling between the incident plane wave and the leaky parallel-plate-like waveguide modes. Specifically, the transmission peak frequencies are accurately predicted.