High-resolution observations of quiescent filaments point out that their fine structure is made of small-scale threads or fibrils. These fibrils can be represented as thin loops having cool tops, i.e. the prominence material, while the rest of the loop displays coronal temperatures. Then, the stacking of these thin loops in the vertical and horizontal directions gives place to the fine structure of filaments. On the other hand, two-dimensional, high-resolution observations of oscillations in filaments suggest that individual fibrils or groups of fibrils may oscillate independently with their own periods (Yi et al. cite{yi}). Díaz et al. (cite{doeb01}, hereafter Paper I) studied the fast magnetohydrodynamic oscillations of a single and isolated slab prominence fibril using a two-dimensional model with no dependence in the y-direction. Here, we introduce a completely different approach which allows us to build up a 3-dimensional model for the fast magnetohydrodynamic oscillations of the fibril configuration used in Paper I. As compared to the results obtained in Paper I, two relevant new features appear; first of all, the cut-off frequency varies with the longitudinal wavenumber, so more modes can be trapped within the fibril; secondly, a much better confinement of the energy of the modes within the fibril appears, diminishing the leakage of energy towards neighbouring fibrils and so difficulting their mutual excitation.