We have investigated the optical properties of two-dimensional plasmonic crystals with triangular lattice structures composed of cylindrical pillars and air holes using rigorous coupled-wave analysis. We revealed the complex band behavior and the radiation properties, which depend on the relative cylinder-radius with respect to the grating pitch, especially around the Γ point. There are always three apparent modes, some of which are degenerate at the Γ point. The number of apparent modes at the Γ point is different from that in photonic crystal slabs with the same lattice structure, which exhibit four apparent modes at that point. All three apparent modes couple to p-polarized radiation, whereas only two of the three apparent modes couple to s-polarized radiation. We also revealed the effect of the shape of the cylinders that compose the lattice of the plasmonic crystals. Deviation of the cylinder cross-section from a perfect circle makes the dispersion relation more complicated and reduces the mode density.