An anisotropic Lagrangian approach is used to investigate the influence of a mean fluid velocity gradient on the motion and on the dispersion of heavy particles suspended in a stationary homogeneous turbulent gas flow. The problem of consistency of the correlation matrices used in first-order stochastic processes is put forward. The accuracy of the method is first checked by comparing the numerical predictions with available experimental results in uniform flow. A second validation test is carried out by comparison with dispersion measurements in a horizontal shear flow. Numerical predictions are then provided in the case of a vertical upward turbulent shear flow, the gravity force held implying a nonzero mean drift velocity between fluid and particles. The transverse particle dispersion is found to be slightly reduced by the presence of a uniform shear, The streamwise particle turbulent intensity is enhanced above the level predicted without shear, whereas, in the direction normal to the flow, the fluctuating velocity of particles are found to be unaffected by the presence of shear, The effect of the lift force due to the mean shear is found to be negligible, except for the mean transverse deviation, which is found however not to be caused only by the lift. The presented numerical results, which confirm and extend the theoretical analysis of others, may explain some experimental observations reported in the literature, (C) 2000 Elsevier Science Ltd. All rights reserved.