We describe a Herschel Space Observatory 194-671 mu m spectroscopic survey of a sample of 121 local luminous infrared galaxies and report the fluxes of the CO J to J-1 rotational transitions for 4 <= J <= 13, the [N II] 205 mu m line, the [C I] lines at 609 and 370 mu m, as well as additional and usually fainter lines. The CO spectral line energy distributions (SLEDs) presented here are consistent with our earlier work, which was based on a smaller sample, that calls for two distinct molecular gas components in general: (i) a cold component, which emits CO lines primarily at J less than or similar to 4 and likely represents the same gas phase traced by CO (1-0), and (ii) a warm component, which dominates over the mid-J regime (4 < J less than or similar to 10) and is intimately related to current star formation. We present evidence that the CO line emission associated with an active galactic nucleus is significant only at J > 10. The flux ratios of the two [C I] lines imply modest excitation temperatures of 15-30 K; the [C I] 370 mu m line scales more linearly in flux with CO (4-3) than with CO (7-6). These findings suggest that the [C I] emission is predominantly associated with the gas component defined in (i) above. Our analysis of the stacked spectra in different far-infrared (FIR) color bins reveals an evolution of the SLED of the rotational transitions of H2O vapor as a function of the FIR color in a direction consistent with infrared photon pumping.