  We determined effective temperatures, surface gravities, and metallicities
for a sample of 343 M dwarfs observed with CARMENES, the double-channel,
high-resolution spectrograph installed at the 3.5 m telescope at Calar Alto
Observatory. We employed SteParSyn, a Bayesian spectral synthesis
implementation particularly designed to infer the stellar atmospheric
parameters of late-type stars following a Markov chain Monte Carlo approach. We
made use of the BT-Settl model atmospheres and the radiative transfer code
turbospectrum to compute a grid of synthetic spectra around 75 magnetically
insensitive Fe I and Ti I lines plus the TiO $\gamma$ and $\epsilon$ bands. To
avoid any potential degeneracy in the parameter space, we imposed Bayesian
priors on Teff and log g based on the comprehensive, multi-band photometric
data available for the sample. We find that this methodology is suitable down
to M7.0 V, where refractory metals such as Ti are expected to condense in the
stellar photospheres. The derived $T_{\rm eff}$, $\log{g}$, and [Fe/H] range
from 3000 to 4200 K, 4.5 to 5.3 dex, and -0.7 to 0.2 dex, respectively.
Although our $T_{\rm eff}$ scale is in good agreement with the literature, we
report large discrepancies in the [Fe/H] scales, which might arise from the
different methodologies and sets of lines considered. However, our [Fe/H] is in
agreement with the metallicity distribution of FGK-type stars in the solar
neighbourhood and correlates well with the kinematic membership of the targets
in the Galactic populations. Lastly, excellent agreement in $T_{\rm eff}$ is
found for M dwarfs with interferometric angular diameter measurements, as well
as in the [Fe/H] between the components in the wide physical FGK+M and M+M
systems included in our sample.
