(Ultra) luminous infrared galaxies ((U)LIRGs) are objects characterized by their extreme infrared (8-1000 mu m) luminosities (L sub(LIRG) > 10 super(11) L sub(middot in circle) and L sub(ULIRG) > 10 super(12)L sub(middot in circle)). The Herschel Comprehensive ULIRG Emission Survey (PI: van der Werf) presents a representative flux-limited sample of 29 (U)LIRGs that spans the full luminosity range of these objects (10 super(11)L sub(middo t in circle) < or =, slant L sub(IR) < or =, slant 10 super(13) L sub(middot in circle)). With the Herschel Space Observatory, we observe CII 157 mu m, OI 63 mu m, and OI 145 mu m line emission with Photodetector Array Camera and Spectrometer, CO J = 4-3 through J = 13-12, CI 370 mu m, and CI 609 mu m with SPIRE, and low-J CO transitions with ground-based telescopes. The CO ladders of the sample are separated into three classes based on their excitation level. In 13 of the galaxies, the OI 63 mu m emission line is self absorbed. Comparing the CO excitation to the InfraRed Astronomical Satellite 60/100 mu m ratio and to far infrared luminosity, we find that the CO excitation is more correlated to the far infrared colors. We present cooling budgets for the galaxies and find fine-structure line flux deficits in the CII, SiII, OI, and CI lines in the objects with the highest far IR fluxes, but do not observe this for CO 4 < or =, slant J sub(upp) < or =, slant 13. In order to study the heating of the molecular gas, we present a combination of three diagnostic quantities to help determine the dominant heating source. Using the CO excitation, the CO J = 1-0 linewidth, and the active galactic nucleus (AGN) contribution, we conclude that galaxies with large CO linewidths always have high-excitation CO ladders, and often low AGN contributions, suggesting that mechanical heating is important.