A semiclassical wavepacket method has been introduced to study diatom-diatom exchange reactions of the type AB+CD-->ABC+D; ABD+C using purely inelastic coordinates. The vibrations of the two reagent molecules are treated quantum mechanically by an exact solution of the time-dependent Schrodinger equation. The rotational motions as well as the relative translational motion of the two molecules are treated classically. The coupling between the quantal and classical degrees of freedom is described within an effective Hamiltonian approach. The method has been illustrated for the exothermic reaction OH+H-2-->H2O+H by computing integral cross sections and rate constants. The computed integral cross sections are in reasonable accord with existing quasiclassical trajectory results, but in significant disagreement with recently reported approximate quantum mechanical calculations. The calculated rate constants for the ground vibrational states of the reagents are in good agreement with experimental rate constants in the temperature range 300 less than or equal to T less than or equal to 700 K. Comparisons of the rate constants are also made with various theoretical results, including a recently reported six-dimensional quantum mechanical calculation.