We develop a systematic way for estimating multidimensional Franck-Condon factors and transition rates for vertical and nonvertical transitions. By analyzing the phase-space overlap integral, we find the most probable positions and momenta of the nuclei immediately after the electronic transition. We find the transition rate by treating the dominant region in phase space as a funnel for the transition and by calculating the flow of probability through this funnel. We use the Wigner representation and its semiclassical limit and find that the transition occurs through a point(s) on the final surface of constant energy where the initial Wigner function is maximal. This dominant contribution is estimated analytically. Results are illustrated for Harmonic, Morse and Poeschl-Teller oscillators. (C) 2003 American Institute of Physics.