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. Since this is a crude model to represent a real fibril, 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. A new array of modes of oscillation, together with their periods and spatial properties, is described, showing several important differences with respect to the properties of modes in Cartesian geometry. Among other conclusions, our results show that all sausage modes (m=0) possess a cutoff frequency, while the fundamental kink and fluting modes (m>0) do not show such a cutoff. In addition, the frequency of these modes is independent of the azimuthal wavenumber (m) and of the fibril thickness for a wide range of values of this parameter, which is an important fact for prominence seismology. Moreover, the spatial structure of the modes below the cutoff frequency is such that in this geometry perturbations are confined in the dense part of the fibril, the leakage of energy toward the coronal medium being very small, which may prevent the excitation of neighboring 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.