  Dynamic and thermal processes regulate the structure of the multi-phase
interstellar medium (ISM), and ultimately establish how galaxies evolve through
star formation. Thus, to constrain ISM models and better understand the
interplay of these processes, it is of great interest to measure the thermal
pressure ($P_{\rm th}$) of the diffuse, neutral gas. By combining [C II] 158
$\mu$m, HI, and CO data from 31 galaxies selected from the Herschel KINGFISH
sample, we have measured thermal pressures in 534 predominantly atomic regions
with typical sizes of $\sim$1 kiloparsec. We find a distribution of thermal
pressures in the $P_{\rm th}/k\sim10^3-10^5$ K cm$^{-3}$ range. For a
sub-sample of regions with conditions similar to those of the diffuse, neutral
gas in the Galactic plane, we find thermal pressures that follow a log-normal
distribution with a median value of $P_{\rm th}/k\approx3600$ K cm$^{-3}$.
These results are consistent with thermal pressure measurements using other
observational methods. We find that $P_{\rm th}$ increases with radiation field
strength and star formation activity, as expected from the close link between
the heating of the gas and the star formation rate. Our thermal pressure
measurements fall in the regime where a two-phase ISM with cold and warm
neutral medium could exist in pressure equilibrium. Finally, we find that the
midplane thermal pressure of the diffuse gas is about $\sim30$% of the vertical
weight of the overlying ISM, consistent with results from hydrodynamical
simulations of self-regulated star formation in galactic disks.
