Observations of quiescent filaments show fine-scale structures, suggesting that they can be composed of small-scale threads or fibrils. Here, we study the oscillations of group of periodic fibrils in a Cartesian geometry using Bloch's theorem. The periodic array of fibrils provides a bridge between the seemingly different oscillations of a finite number of fibrils and the oscillations of an homogeneous prominence. Our results show that fine structure plays an important role and cannot be neglected. The filling factor can be used to obtain reliable estimates of periods. Also, the presence of the fibrils introduces a fine-scale modulation in the spatial structure of the global oscillations.