In this paper we present fluxes in the [C I] lines of neutral carbon at the centers of some 76 galaxies with far-infrared luminosities ranging from 10 9 to 10 12 L , as obtained with the Herschel Space Observatory and ground-based facilities, along with the line fluxes of the J = 7−6, J = 4−3, J = 2−1 12 CO, and J = 2−1 13 CO transitions. With this dataset, we determine the behavior of the observed lines with respect to each other and then investigate whether they can be used to characterize the molecular interstellar medium (ISM) of the parent galaxies in simple ways and how the molecular gas properties define the model results. In most starburst galaxies, the [C I] to 13 CO line flux ratio is much higher than in Galactic star-forming regions, and it is correlated to the total far-infrared luminosity. The [C I] (1−0)/ 12 CO (4−3), the [C I] (2−1)/ 12 CO (7−6), and the [C I] (2−1)/(1−0) flux ratios are correlated, and they trace the excitation of the molecular gas. In the most luminous infrared galaxies (LIRGs), the ISM is fully dominated by dense (n( H 2 ) = 10 4 −10 5 cm −3 ) and moderately warm (T kin ≈ 30 K) gas clouds that appear to have low [C • ]/[CO] and [ 13 CO]/[ 12 CO] abundances. In less luminous galaxies, emission from gas clouds at lower densities becomes progressively more important, and a multiple-phase analysis is required to determine consistent physical characteristics. Neither the 12 CO nor the [C I] velocity-integrated line fluxes are good predictors of molecular hydrogen column densities in individual galaxies. In particular, so-called X([C I]) conversion factors are not superior to X( 12 CO) factors. The methods and diagnostic diagrams outlined in this paper also provide a new and relatively straightforward means of deriving the physical characteristics of molecular gas in high-redshift galaxies up to z = 5, which are otherwise hard to determine.