We calculate the turbulent dissipation rate of incompressive hydromagnetic fluctuations and the resulting radial evolution of temperature in the solar wind using generalizations of Kolmogorov and Kraichnan MHD turbulence phenomenologies that include the suppression of dissipation by high cross helicity. The results for the temperature evolution are compared to a variety of data sets to test the phenomenologies over a wide parameter range. Motivated by the observations, we use different power laws in radius for the amplitudes of Alfv¿nic and nonAlfv¿nic fluctuations to determine the cascade rates. To explain the observations using Kolmogorov-like models, we found it necessary to suppress the dissipation rates for the high cross helicity streams even further than predicted by simple models; this may be due to the nonequilibrium nature of the spectrum or to other causes as yet unknown. The Kolmogorov-like model gives rise to a significant amount of turbulent heating, implying that turbulent heating, while likely dominant only in the inner heliopshere, may be competitive with heating by shocks and the assimilation of interstellar pickup ions in the outer heliosphere. In contrast, the generalized Kraichnan phenomenology yields less turbulent heating than the Kolmogorov-like model and seems inadequate to explain the observations. We conclude that while no existing turbulence model adequately explains the observed radial dependence of temperature in the wind, solar there appears to be sufficient energy available for turbulent heating to contribute significantly, even in the outer heliosphere. ¿American Geophysical Union 1995