A self-contained formulation for analyzing electromagnetic scattering by a significant class of planar gratings composed of plasmonic nanorods, which were infinite length along their axes, is presented. The procedure for the lattice sums technique was implemented in a cylindrical harmonic expansion method based on the generalized reflection matrix approach for full-wave scattering analysis of plasmonic gratings. The method provided a high computational efficiency and can be considered as one of the best-suited numerical tools for the optimization of plasmonic sensors and plasmonic guiding devices both having a planar geometry. Although the proposed formalism can be applied to analyze a wide class of plasmonic gratings, three configurations were studied in the manuscript. Firstly, a multilayered grating of silver nanocylinders formed analogously to photonic crystals was considered. In the region far from the resonances of a single plasmonic nanocylinder, the structure showed similar properties compared to conventional photonic crystals. When one or a few nanorods were periodically removed from the original crystal, thus forming a crystal with defects, a new band was formed in the spectral responses because of the resonant tunneling through the defect layers. The rigorous formulation of plasmonic gratings with defects was proposed for the first time. Finally, a plasmonic planar grating of metal-coated dielectric nanorods coupled to the dielectric slab was investigated from the viewpoint of design of a refractive index sensor. Dual-absorption bands attributable to the excitation of the localized surface plasmons were studied, and the near field distributions were given in both absorption bands associated with the resonances on the upper and inner surfaces of a single metal-coated nanocylinder. Resonance in the second absorption band was sensitive to the refractive index of the background medium and could be useful for the design of refractive index sensors. Also analyzed was a phase-matching condition between the evanescent space-harmonics of the plasmonic grating and the guided modes inside the slab, leading to a strong coupling.