Split ring resonator arrays are difficult to achieve miniaturization because of their big sizes, while complementary split ring resonators are easy to damage the other parts of the microwave circuit because it is etched on the ground layer. In order to solve these problems, a novel double-layer dielectric composite right/left-handed transmission line structure is proposed and its equivalent relative permittivity and permeability are calculated. Then the influence of dimension parameters of this novel structure on transmission characteristics is analyzed. Finally two dual-band bandpass filters are designed and fabricaled by applying a new Minkowski fractal curve to this structure, and meanwhile the dual-band characteristic of this composite right/left transmission line is demonstrated.