We report high-resolution, picosecond laser spectroscopy measurements of the relaxation of lowest-energy heavy-hole excitons in GaAs quantum wells grown with and without growth interruptions. In both cases excitons relax by losing potential energy in a diffusion motion driven by potential fluctuations in the quantum-well plane. In the quantum wells grown without growth interruptions the low-energy shift of exciton lines is comparable with the width of an inhomogeneously broadened line and shows two-step decay with a slow exponential component characteristic for exciton localization at well interfaces. In the growth-interrupted quantum wells in which the size of islands with constant well thickness is large compared with exciton diameter we observe splitting of the heavy-hole transitions into the multiplets of narrow lines corresponding to one monolayer difference in the well width. The energy shifts of each line in this case amount to only a fraction of the width of individual lines (or there is no shift at all) suggesting the interisland migration of excitons mediated by acoustic phonon scattering as being responsible for exciton relaxation. Again a two-step decay of the luminescence is observed at low temperatures (2 K). Temperature-dependent measurements show that at higher temperatures luminescence decay becomes governed by a single exponential as expected for delocalized excitons (no energy shift is observed during the exciton decay time). This allowed us to study directly intrinsic properties of excitons in quantum wells, i.e, to determine the lifetime of K-parallel to=0 excitons, which is a fundamental parameter of the system and has been assessed by many theories. The experimentally determined, from the present work, values of the radiative lifetime an 24.4 ps for a 13 ML thick well and 21.8 ps for a 17 ML well. These values agree very well with theoretical estimates of Andreani ct al. (Solid State Cornmun., 77 (1991) 641). The effective lifetimes measured as a time to decay to lie of the value of the maximum of PL intensity are considerably longer than the radiative lifetime, since in thermal equilibrium only a small fraction of excitons occupy the states with k(1) < k(0) which can decay radiatively.