We present optical spectroscopy of galaxies in clusters detected through the Sunyaev-Zel'dovich (SZ) effect with the South Pole Telescope (SPT). We report our own measurements of 61 spectroscopic cluster redshifts, and 48 velocity dispersions each calculated with more than 15 member galaxies. This catalog also includes 19 dispersions of SPT-observed clusters previously reported in the literature. The majority of the clusters in this paper are SPT-discovered; of these, most have been previously reported in other SPT cluster catalogs, and five are reported here as SPT discoveries for the first time. By performing a resampling analysis of galaxy velocities, we find that unbiased velocity dispersions can be obtained from a relatively small number of member galaxies (≾30), but with increased systematic scatter. We use this analysis to determine statistical confidence intervals that include the effect of membership selection. We fit scaling relations between the observed cluster velocity dispersions and mass estimates from SZ and X-ray observables. In both cases, the results are consistent with the scaling relation between velocity dispersion and mass expected from dark-matter simulations. We measure a ~30% log-normal scatter in dispersion at fixed mass, and a ~10% offset in the normalization of the dispersion-mass relation when compared to the expectation from simulations, which is within the expected level of systematic uncertainty.