Spectroscopic observations are presented for objects in the fields of 4 galaxy clusters at z=0.7-0.8 (Cl 1040.7-1155 (Cl 1040), Cl 1054.4-1146 (Cl 1054-11), Cl 1054.7-1245 (Cl 1054-12), and Cl 1216.8-1201 (Cl 1216)) and in one cluster at z~0.5 (Cl 1232.5-1250 (Cl 1232)) of the ESO Distant Cluster Survey (EDisCS). We present tables of RA and DE positions for an equinox of 2000 (J2000, epoch 2001), I-band magnitudes for an aperture of radius 1 arcsecond, spectroscopic redshifts for objects in the fields of our 5 clusters, and membership and targeting flags to indicate whether the object is a cluster member and/or was targeted to be a cluster member respectively.