Some observations suggest that quiescent solar prominences can be considered as composed by small-scale loops, or fibrils, which are stacked one after another in both the vertical and horizontal directions. In a previous work we studied, in Cartesian geometry, the propagation of fast MHD waves in a two-dimensional magnetostatic model representing one of these fibrils. In this paper we use a more realistic model based on a cylindrically symmetric flux tube and study the propagation of fast, MHD waves in this structure. Among other conclusions, our results show that all sausage modes (m = 0) possess a cut-off frequency, while the fundamental kink and fluting modes (m > 0) do not show such a cut-off. Moreover, the spatial structure of the modes below the cut-off frequency is such that in this geometry perturbations are confined in the dense part of the fibril, the leakage of energy towards the coronal medium being very small, which may prevent the excitation of neighbouring fibrils. Finally, diagnostic diagrams displaying the oscillatory period in terms of some equilibrium parameters are provided in order to allow for a comparison between our theoretical results and those coming from observations.