A method of calculation of rotation-vibration states for a general triatomic that places the body-fixed z axis perpendicular to the plane of the molecule is implemented within a discrete variable representation (DVR) for the vibrational motion. Calculations are presented for water and H-3(+). For H-3(+) the new method improves on previous high accuracy ab initio treatments of the rotation-vibration energies of the molecule both in accuracy and the range of rotational states that can be treated. Reliable treatment of quasilinear geometries means that the method is also promising for treating very highly excited states.