We investigate the stellar kinematics and stellar populations of 58 radio-loud galaxies of intermediate luminosities (L-3 GHz > 10(23) W Hz(-1)) at 0.6 < z < 1. This sample is constructed by cross-matching galaxies from the deep VLT/VIMOS LEGA-C spectroscopic survey with the VLA 3 GHz data set. The LEGA-C continuum spectra reveal for the first time stellar velocity dispersions and age indicators of z similar to 1 radio galaxies. We find that z similar to 1 radio-loud active galactic nucleus (AGN) occur exclusively in predominantly old galaxies with high velocity dispersions: sigma(*) > 175 kms(-1), corresponding to black hole masses in excess of 10(8) M-circle dot. Furthermore, we confirm that at a fixed stellar mass the fraction of radio-loud AGN at z similar to 1 is five to 10 times higher than in the local universe, suggesting that quiescent, massive galaxies at z similar to 1 switch on as radio AGN on average once every Gyr. Our results strengthen the existing evidence for a link between high black hole masses, radio loudness, and quiescence at z similar to 1.