A full variational procedure is presented for the calculation of rovibrational (J>O) energy levels which is particularly suited for triatomic potentials that support large amplitude motions and that may be of high permutational symmetry. It is based on a kinetic energy operator expressed in hyperspherical coordinates (p,theta,Phi). Particular attention is paid to the singularities of this operator in the derivation of the primitive basis functions, which should exactly cancel all singularities, and in their subsequent contractions. The method is applied to the D-3h molecules H-3(+) and Na-3(+), for which converged rovibrational energies are calculated for J=O, 1, 2 to 25 000 (-1) for H-3(+) and to 1250 cm(-1) for Na-3(+), respectively. A spectral analysis of these energy levels cm is undertaken. For the lowest ten vibrational levels of H-3(+), converged rovibrational energies up to J=10 are also calculated. These energies, which extend previous calculations, should prove useful in the interpretation of the observed spectra. Our J= 1 results for H-3(+) compare well with recently published values. The results for Na-3(+) constitute new data which supplement our previous J=O calculations. It is shown that the spectrum of this molecule can be expressed very well by an appropriate effective Hamiltonian, quite in contrast to H-3(+). In order to verify that the method is general, J>O calculations are performed for model potentials of H2O+ (C-2V) and HLiH- (D infinity h). The results are in full agreement with those from a proven variational method in valence coordinates.