The frequency selective characteristics of a new dielectric periodic structure composed of left handed materials were carefully studied by a method which combines the multimode network theory with the rigorous mode matching method. The emphasis was laid on the investigation of the variations of the frequency selective characteristics with structure parameters. The comparisons for the frequency selective properties between RHM and LHM gratings were also shown with some explanations. It is indicated that the bandwidth of the total reflection spectrum for the LHM gratings is much larger than that for the RHM ones. The calculated results are of significance for accurate design of the new millimeter wave frequency selective surface.