A three-dimensional semiclassical analytic model of vibrational energy transfer in collisions between a rotating diatomic molecule and an atom has been developed. The model is based on analysis of classical trajectories of a free-rotating (FR) molecule acted upon by a superposition of repulsive exponential atom-to-atom potentials. The energy transfer probabilities have been evaluated using the nonperturbative forced harmonic oscillator (FHO) model. The model predicts the probabilities for vibrational energy transfer as functions of the total collision energy, orientation of a molecule during a collision, its rotational energy, and impact parameter. The model predictions have been compared with the results of three-dimensional close-coupled semiclassical trajectory calculations using the same potential-energy surface. The comparison demonstrates not only remarkably good agreement between the analytic and numerical probabilities across a wide range of collision energies, but also shows that the analytic FHO-FR model correctly reproduces the probability dependence on other collision parameters such as rotation angle, angular momentum angle, rotational energy, impact parameter, and collision reduced mass. The model equally well predicts the cross sections of single-quantum and multiquantum transitions and is applicable up to very high-collision energies and quantum numbers. Most importantly, the resultant analytic expressions for the probabilities do not contain any arbitrary adjustable parameters commonly referred to as "steric factors." The model provides new insight into kinetics of vibrational energy transfer and yields accurate expressions for energy-transfer rates that can be used in kinetic modeling calculations.