This paper describes the use of an exact fast Fourier transform method to prepare specified vibrational-rotational states of triatomic molecules. The method determines the Fourier coefficients needed to describe the coordinates and momenta of a vibrating-rotating triatomic molecule. Once the Fourier coefficients of a particular state are determined, it is possible to easily generate as many random sets of initial Cartesian coordinates and momenta as desired. All the members of each set will correspond to the particular vibrational-rotational state selected. For example, in the case of the ground vibrational state of a nonrotating water molecule, the calculated actions of 100 sets of initial conditions produced actions within 0.001HBAR of the specified quantization values and energies within 5 cm(-1) of the semiclassical eigenvalue. The numerical procedure is straightforward for states in which all the fundamental frequencies are independent. However, for states for which the fundamental frequencies become commensurate (resonance states), there are additional complications. In these cases it is necessary to determine a new set of "fundamental" frequencies and to modify the quantization conditions. Once these adjustments are made, good results are obtained for resonance states. The major problems are in labeling the large number of Fourier coefficients and the presence of regions of chaotic motion. Results are presented for the vibrational states of H2O and HCN and the rovibrational states of H2O.