Redshifts for 23 possible radio galaxies with radio magnitudes of less than 17 in the field of the Coma cluster are reported and discussed. Of the 16 most likely cluster members, half are nonelliptical galaxies with emission lines in their spectra and the remainder are non-emission-line E or S0 galaxies. Except for two of the weakest sources, the galaxies have redshifts which average 600 km/s greater than the cluster mean. The excess redshift does not appear to depend upon emission-line properties or distance from the cluster center. The non-emission-line galaxies are brighter both optically and at radio wavelengths than the emission-line objects. A direct correlation of radio magnitude and redshift appears to exist for the non-emission-line radio sources.