  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
($\lesssim 30$), 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
$\sim$30% log-normal scatter in dispersion at fixed mass, and a $\sim$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.
