Considering epicyclic oscillations of pressure-supported perfect fluid tori orbiting Kerr black holes we examine non-geodesic (pressure) effects on the epicyclic modes properties. Using a perturbation method we derive fully general relativistic formulae for eigenfunctions and eigenfrequencies of the radial and vertical epicyclic modes of a slightly non-slender, constant specific angular momentum torus up to second-order accuracy with respect to the torus thickness. The behaviour of the axisymmetric and lowest-order (m = ±1) non-axisymmetric epicyclic modes is investigated. For an arbitrary black hole spin we find that, in comparison with the (axisymmetric) epicyclic frequencies of free test particles, non-slender tori receive negative pressure corrections and thus exhibit lower frequencies. Our findings are in qualitative agreement with the results of a recent pseudo-Newtonian study of the analogous problem defined within the Paczyński–Wiita potential. Implications of our results on the high-frequency quasi-periodic oscillation (QPO) models dealing with epicyclic oscillations are addressed.