The influences of the wave vectors off the periodic plane on the distribution of energy bands of triangular, honeycomb and square lattices are analyzed by plane wave method. The rods and background of these lattices are all composed of two mediums with dielectric constants equal to 13 and 1. With the increase of wave vectors off periodic plane, many changes will take place in the dispersion relation of these lattices: the band gaps of wave vectors in the periodic plane will decrease gradually and even disappear; a region, which does not exist modes, will appear in the low frequency range and it will be broaden with the increase of wave vectors off periodic plane; new degenerated energy levels appear and primarily degenerated levels will disappear; the energy bands tend to become flat; the absolute band gaps can easily form among the lowest energy bands. By the analyses of the band gaps of three lattices, some conclusions can be obtained: the width of absolute band gaps in triangular and honeycomb lattices are larger than those of square lattices, which make triangular and honeycomb lattices lower the rate of spontaneous radiation more effectively than square lattices; the width of band gaps of triangular lattices in the leaky region is larger than that of honeycomb lattices, which made triangular lattices be more appropriate to be used as reflect mirror than honeycomb lattices.