A simple kinematic model based on the concept of an orientation-dependent critical configuration for reaction is introduced and applied. The model serves two complementary purposes. The very concept of a critical configuration of no return provides a practical link between the experimental laboratory-based measurements and the features of the potential energy surface. It thus provides a route from the observations to the orientation dependence of the barrier for reaction. In the predictive mode the model provides an easily implemented procedure for computing the reactivity of oriented reagents (including those actually amenable to measurement) from a given potential energy surface. The predictions of the model are compared against classical trajectory results for the H + D2 reaction. By use of realistic potential energy surfaces the model is applied to the Li + HF and O + HCl reactions where the HX molecules are pumped by a polarized laser. A given classical trajectory is deemed reactive or not according to whether it can surmount the barrier at that particular orientation. The essential difference with the model of Levine and Bernstein is that the averaging over initial conditions is performed by using a Monte Carlo integration. One can therefore use the correct orientation-dependent shape (and not only height) of the barrier to reaction and, furthermore, use oriented or aligned reagents. Since the only numerical step is a Monte Carlo sampling of initial conditions, very many trajectories can be "run". This suffices to determine the reaction cross section for different initial conditions. To probe the products, we have employed the kinematic approach of Elsum and Gordon. The result is a model where, under varying initial conditions, examining final-state distributions or screening different potential energy surfaces can be efficiently carried out.