Abstract-We have calculated the photonic bands of a dispersive and lossy periodic array of left-handed metamaterial layers in air. Depending on the behavior of the fields inside the metamaterial component, two categories of modes for oblique propagation are identified: the oscillatory and the tunneling modes. In order to characterize these two types of solutions, we calculate the complex photonic bands; a criterion of penetration-limit is introduced to quantify the absorption effects. Our results show that oscillatory TE and TM waves can be excited by light incident from air at low frequencies (within the metamaterial regime). In the region of high frequencies only TE tunneling modes are available. To complement the description of the absorption effects, we present transmission spectra and field profiles for TE waves in finite layered systems for the two types of modes here studied.