We present a three-dimensional cloud model designed for the study of formation and evolution of moist convective storms on the giant planets. This is a finite-difference model that solves the dynamic and thermodynamic equations forward in time under the anelastic approximation including microphysics in a parameterized form. Three-dimensional modeling is needed to include the dynamical effects of the vertical and meridional ambient wind shears and the Coriolis force on the storm onset and evolution. The model is designed to run on the four giant planets to compare how moist convection operates in each case. In this paper we present the results for Jupiter water-ammonia storms under a variety of conditions. Several improvements in the mathematical formulation and in the numerical methods and an update of the input values (e.g., the thermal profile and gas abundances) with respect to previous models have been incorporated. We show that under the appropriate relative humidity conditions and particulate precipitation, it is possible to create visible storms even with a small deep water abundance of ∼0.2 times the solar value. The effect of the vertical wind shear is important, as it is on Earth, in controlling the structure of the cloud, its growing ratio, and the local wind field; but it has a small influence on the upper level reached by the storm. The distribution of the convective cells created in the presence of a vertical wind shear resembles the pattern of mesoscale storms recently observed by Galileo.