We have investigated the influence of the electronic band structure on the exciton dynamics in GaAsP tensile-strained quantum wells. We have found that the exciton cooling time is notably reduced when the heavy- and light-hole excitons are degenerate. The lifetime of the heavy-hole exciton is ~300 ps whereas it is ~500 ps for the light-hole exciton. Furthermore, we have determined, from the initial degree of polarization of the emission, the valence-band mixing as a function of the energy splitting between the heavy-hole and light-hole subbands. The degree of mixing is in qualitative agreement with tight-binding calculations.