We analyze transmission of electromagnetic waves through a one-dimensional periodic layered structure consisting of slabs of a left-handed metamaterial and air. We derive the effective parameters of the metamaterial from a microscopic structure of wires and split-ring resonators possessing the left-handed characteristics in the microwave frequency range, and then study, by means of the transfer-matrix approach and the finite-difference time-domain numerical simulations, the transmission properties of this layered structure in a band gap associated with the zero averaged refractive index. By introducing defects, the transmission of such a structure can be made tunable, and we study the similarities and differences of the defects modes excited in two types of the band gaps.