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A novel periodic structure, made of an arbitrary number of stacked subwavelength hole arrays, exhibiting simultaneously electromagnetic band gap, extraordinary transmission and a longitudinal left handed propagation is presented in this paper. If the longitudinal period of the stacked structure is chosen adequately, it is possible under normal incidence to mold the electromagnetic wave properties inside the structure from right-handed to left-handed wave propagation passing through a zero-group velocity band. The transmission response of the fabricated prototype has been measured with a millimeter wave quasioptical vector network analyzer in the range between 40 GHz and 110 GHz confirming the possibility to tune the left- or right-handed characteristics of the propagating waves. These results can give rise to interesting applications such as novel lenses and other quasioptical structures.