Mid-infrared molecular hydrogen (H-2) emission is a powerful cooling agent in galaxy mergers and in radio galaxies; it is a potential key tracer of gas evolution and energy dissipation associated with mergers, star formation, and accretion onto supermassive black holes. We detect mid-IR H-2 line emission in at least one rotational transition in 91% of the 214 Luminous Infrared Galaxies (LIRGs) observed with Spitzer as part of the Great Observatories All-sky LIRG Survey. We use H-2 excitation diagrams to estimate the range of masses and temperatures of warm molecular gas in these galaxies. We find that LIRGs in which the IR emission originates mostly from the Active Galactic Nuclei (AGN) have about 100 K higher H-2 mass-averaged excitation temperatures than LIRGs in which the IR emission originates mostly from star formation. Between 10% and 15% of LIRGs have H-2 emission lines that are sufficiently broad to be resolved or partially resolved by the high-resolution modules of Spitzer's Infrared Spectrograph (IRS). Those sources tend to be mergers and contain AGN. This suggests that a significant fraction of the H-2 line emission is powered by AGN activity through X-rays, cosmic rays, and turbulence. We find a statistically significant correlation between the kinetic energy in the H-2 gas and the H-2 to IR luminosity ratio. The sources with the largest warm gas kinetic energies are mergers. We speculate that mergers increase the production of bulk inflows leading to observable broad H-2 profiles and possibly denser gas.