We present new radial velocities for 87 globular clusters around the elliptical galaxy NGC 4472 and combine these with our previously published data to create a data set of velocities for 144 globular clusters around NGC 4472. We utilize this data set to analyze the kinematics of the NGC 4472 globular cluster system. The new data confirm our previous discovery that the metal-poor clusters have significantly higher velocity dispersion than the metal-rich clusters in NGC 4472. We also find very little or no rotation in the more spatially concentrated metal-rich population, with a resulting upper limit for this population of (v/σ)proj<0.34 at a 99% confidence level. The very small angular momentum in the metal-rich population requires efficient angular momentum transport during the formation of this population, which is spatially concentrated and chemically enriched. Such angular momentum transfer can be provided by galaxy mergers, but it has not been achieved in other extant models of elliptical galaxy formation that include dark matter halos. We also calculate the velocity dispersion as a function of radius and show that it is consistent with roughly isotropic orbits for the clusters and the mass distribution of NGC 4472 inferred from X-ray observations of the hot gas around the galaxy.