A three-component quasiperiodic superlattice structures composing of both positive and negative refractive index materials are shown to display resonant transport behavior and optical band gaps. When the structure is composed of nondispersive refractive index material, the number of the resonant transmission peaks increases and the optical band gap becomes broad with the increasing of the medium generation. The band gap covers all the wavelength except for some singular wavelength points when the structure is composed of negative refractive index materials. Moreover, it is found that the spectrum shifts to low frequency for oblique incidence. And with the increasing of the optical thickness, the band gap splits and new perfect transport channels emerge. For a more realistic dispersive negative refractive index material, the transmission coefficients are characterized by a rich transmission profile without symmetry, more wide band gaps and abundance transmissive channels appear.