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 $P_{rm th}/ksim10^3-10^5$ 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 $P_{rm th}/kapprox3600$ 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.
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