The propagation of electromagnetic waves through a 1-dimensional layered system containing alternate layers of air and a uniaxial, anisotropic, left-handed material is investigated. The optical axis of this material is along the stacking direction and the components of the electric permittivity and magnetic permeability tensors that characterize the metamaterial are described by Drude-type responses. Different plasmon frequencies are considered for directions parallel and perpendicular to the optical axis. As in the isotropic case, plasmon polariton modes are found in the neighborhood of the plasmon frequency corresponding to the optical axis. Moreover, it is shown that, depending on the relation between the two plasmon frequencies of the metamaterial, anisotropy leads to the unfolding of an infinite number of nearly dispersionless plasmon-polariton bands either above or below the parallel plasmon frequency.