  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. We aim to determine oscillation
properties of fibrils not only in the plane-of-the-sky (horizontal) direction,
but also along the line-of-sight (vertical) direction. We obtained
imaging-spectroscopy data in Fe I 6173, Ca II 8542, and Ca II K obtained with
the Swedish 1-m Solar Telescope. We created a sample of 120 bright Ca II K
fibrils and measured their horizontal motions. Their vertical motion was
determined through 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. The
bright Ca II K fibrils carry transverse waves with a mean period of
$2.2\times10^2$ s, and a horizontal velocity amplitude of 2 km s$^{-1}$,
consistent with earlier results. The mean vertical velocity amplitude is 1 km
s$^{-1}$. We find that 118 out of the 120 fibrils carry waves in both the
vertical and horizontal directions, and 55 of those have identical periods. For
those 55, we find that all phase differences between $0$ and $2\pi$ occur, with
a mild but significant preference for linearly polarized waves (phase
difference of $0$ or $\pi$). 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 on imaging
data only are too low because they ignore the contribution of the vertical
velocity.
