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In this paper, we present two different approaches to the problem of wave propagation in two-dimensional (2-D) periodic structures; one is the search of dispersion roots of which the real and imaginary parts represent the phase and decay constants of an unbound 2-D periodic medium, respectively, and the other is the investigation of the scattering characteristics of a finite 2-D periodic structure that is treated as a stack of 1-D periodic layers. Specifically, the rigorous mode-matching method is employed for both approaches, and the class of 2-D periodic structures with metal rods of rectangular cross-section is considered explicitly for mathematical formulation and quantitative analysis of associated physical phenomena. The mutual verifications of the results by the two different approaches facilitate the understanding of band structure of the 2-D periodic medium. In addition to the stopbands that can be easily identified to be due to the individual periodicity in either x or y direction, particular attention is directed to the combined effect of both periodicities, which results in the extra stopbands that are slanted at an angle on the Brillouin diagram. This provides the physical basis for the explanation of the unusually strong reflection of an incident plane wave in certain range of frequency or incident angle.