This paper is an extension of recent work by Isenberg and Hollweg dealing with the preferential acceleration and heating of solar wind heavy ions by a resonant cyclotron interaction with a turbulent spectrum of ion-cyclotron waves. Two major approximations of the previous work are eliminated: that of isotropic ion distributions, and that of nondispersive waves. The wave action flux of finite-amplitude Alfv¿n waves in a multi-ion, differentially flowing, gyrotropic plasma is derived. This quantity is incorporated into the wave-driven solar wind model of Isenberg and Hollweg, which is then applied to gyrotropic protons and alpha particles. It is found that allowing for nonisotropic ions yields enhanced preferential acceleration of alpha particles. However, when dispersion is included in the resonant interaction, the preferential effects are again reduced. These more realistic calculations show that the observations of solar wind alpha particles cannot be explained by a resonant cyclotron interaction.