We investigate the nature of the continuum emission and stellar populations in the inner 1–3 kpc of a complete sample of 24 southern radio galaxies, and we compare the results with a control sample of 18 non-active early-type galaxies. 12 of the radio galaxies are classified as Fanaroff–Riley type I (FR I), eight as FR II and four as intermediate or undefined type (FR x). Optical long-slit spectra are used to perform spectral synthesis as a function of distance from the nucleus at an average sampling of 0.5–1.0 kpc and to quantify the relative contributions of a blue featureless continuum and stellar population components of different ages. Our main finding is a systematic difference between the stellar populations of the radio and control sample galaxies: the former have a larger contribution from an intermediate-age (1 Gyr) component, suggesting a connection between the present radio activity and a starburst which occurred ∼1 Gyr ago. In addition, we find a correlation between the contribution of the 1-Gyr component and the radio power, suggesting that more massive starbursts have led to more powerful radio emission. A similar relation is found between the radio power and the mean age of the stellar population, in the sense that stronger nuclear activity is found in younger galaxies. We also find that the stellar populations of FR I galaxies are, on average, older and more homogeneous than those of FR IIs. Significant population gradients were found in only four radio galaxies, which are also those with more than 10 per cent of their total flux at 4020 A contributed by age components younger than 100 Myr and/or a featureless continuum (indistinguishable from a 3-Myr-old stellar population).