This is the second of a series of papers investigating the oscillation
properties of relativistic, non-selfgravitating tori orbiting around black
holes. Extending the work done in a Schwarzschild background, we here consider the axisymmetric oscillations of vertically integrated tori in a Kerr
spacetime. The tori are modeled with a number of different non-Keplerian
distributions of specific angular momentum and we discuss how the oscillation
properties depend on these and on the rotation of the central black hole. We
first consider a local analysis to highlight the relations between acoustic and
epicyclic oscillations in a Kerr spacetime and subsequently perform a global
eigenmode analysis to compute the axisymmetric p modes. In analogy with what found in a Schwarzschild background, these modes behave as sound waves that are modified by rotation and are globally trapped in the torus. For constant distributions of specific angular momentum, the eigenfrequencies appear in a sequence 2:3:4:... which is essentially independent of the size of the disc and of the black hole rotation. For non-constant distributions of angular momentum, on the other hand, the sequence depends on the properties of the disc and on the spin of the black hole, becoming harmonic for sufficiently large tori. We also comment on how p modes could explain the high frequency quasi-periodic oscillations observed in low-mass X-ray binaries with a black hole candidate and the properties of an equivalent model in Newtonian physics.