Aims. We present spectroscopic observations of galaxies in 15 survey fields as part of the ESO Distant Cluster Survey (EDisCS). We determine the redshifts and velocity dispersions of the galaxy clusters located in these fields, and we test for possible substructure in the clusters. Methods. We obtained multi-object mask spectroscopy using the FORS2 instrument at the VLT. We reduced the data with particular attention to the sky subtraction. We implemented the method of Kelson for performing sky subtraction prior to any rebinning/interpolation of the data. From the measured galaxy redshifts, we determine cluster velocity dispersions using the biweight estimator and test for possible substructure in the clusters using the Dressler-Shectman test. Results. The method of subtracting the sky prior to any rebinning/interpolation of the data delivers photon-noise-limited results, whereas the traditional method of subtracting the sky after the data have been rebinned/interpolated results in substantially larger noise for spectra from tilted slits. Redshifts for individual galaxies are presented and redshifts and velocity dispersions are presented for 21 galaxy clusters. For the 9 clusters with at least 20 spectroscopically confirmed members, we present the statistical significance of the presence of substructure obtained from the Dressler-Shectman test, and substructure is detected in two of the clusters. Conclusions. Together with data from our previous paper, spectroscopy and spectroscopic velocity dispersions are now available for 26 EDisCS clusters with redshifts in the range 0.40-0.96 and velocity dispersions in the range $166\,\mathrm{km}\,\mathrm{s}^{-1}{-}1080\,\mathrm{km}\,\mathrm{s}^{-1}$.