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 [NII] 205{mu}m line, the [CI] 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<~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<~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 [CI] lines imply modest excitation temperatures of 15-30K; the [CI] 370{mu}m line scales more linearly in flux with CO (4-3) than with CO (7-6). These findings suggest that the [CI] 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 H_2_O vapor as a function of the FIR color in a direction consistent with infrared photon pumping.