Context. Fibrils in the solar chromosphere carry transverse oscillations as determined from non-spectroscopic imaging data. They are estimated to carry an energy flux of several kW m‑2, which is a significant fraction of the average chromospheric radiative energy losses. Aims. We aim to determine the oscillation properties of fibrils not only in the plane-of-the-sky (horizontal) direction, but also along the line-of-sight (vertical) direction. Methods. We obtained imaging-spectroscopy data in Fe I 6173 Å, Ca II 8542 Å, and Ca II K with the Swedish 1-m Solar Telescope. We created a sample of 605 bright Ca II K fibrils and measured their horizontal motions. Their vertical motion was determined through non-local thermodynamic equilibrium (non-LTE) inversion of the observed spectra. We determined the periods and velocity amplitudes of the fibril oscillations, as well as phase differences between vertical and horizontal oscillations in the fibrils. Results. The bright Ca II K fibrils carry transverse waves with a mean period of 2.1 × 102 s, and a horizontal velocity amplitude of 1 km s‑1, consistent with earlier results. The mean vertical velocity amplitude is 1.1 km s‑1. We find that 77% of the fibrils carry waves in both the vertical and horizontal directions, and 80% of this subsample exhibit oscillations with similar periods in both horizontal and vertical directions. For the latter, we find that all phase differences between 0 and 2π occur with a mild but significant preference for linearly polarised waves (a phase difference of 0 or π). Conclusions. The results are consistent with the scenario where transverse waves are excited by granular buffeting at the photospheric footpoints of the fibrils. Estimates of transverse wave flux based only on imaging data are too low because they ignore the contribution of the vertical velocity.