We recently built the G4Jy-3CRE catalog of extragalactic radio sources. This catalog lists 264 powerful radio sources selected with similar criteria to those of the revised Third Cambridge Catalog, but visible from the Southern Hemisphere. A literature search revealed that 119 sources in the G4Jy-3CRE catalog (i.e., 45%) lack a firm spectroscopic redshift measurement. Here, we present a campaign aimed at acquiring optical spectra of G4Jy-3CRE sources and measuring their redshifts. We used single-slit observations obtained with the Victor Blanco Telescope, the New Technology Telescope, the Southern Astrophysical Research Telescope, and the 2.1 m telescope of the Observatorio Astronomico Nacional at San Pedro Martir, Mexico. In addition, we analyzed Very Large Telescope/MUSE archival observations. From these observations, we report the spectra and redshifts of 93 sources, 42 of which are the first optical spectra and redshift determinations for the respective sources. With our new data, approximately 71% of the sources in the G4Jy-3CRE catalog now have firm spectroscopic redshift measurements. This data set will be the basis of our future analysis of the optical properties of the G4Jy-3CRE catalog.