We relax the horizontal mean wind strongly towards a simple unidirectional linear vertical wind profile in the troposphere. The strength of the shear is varied as a control parameter. We prescribe surface enthalpy fluxes to exclude complications from air-sea interaction. Our results indicate two distinct flow regimes: for weak wind shear, time-averaged rainfall decreases with shear and convection remains disorganized; for larger wind shear, rainfall increases as convection becomes organized into linear mesoscale systems. This non-monotonic dependence of rainfall on shear is observed when the imposed surface fluxes are moderate. For larger surface fluxes, convection in the unsheared basic state is already organized, so increasing wind shear only leads to increasing rainfall. In addition to surface rainfall, we will also discuss the impact of shear on other variables, such as the parameterized large-scale vertical velocity, convective heating and drying, mass fluxes, cloud fraction, and momentum transport.