We have systematically measured the room-temperature thermal conductivity of epitaxial layers of InxGa1-xN alloys with 15 different Indium compositions ranging from 0.08 to 0.98 by time-domain thermoreflectance method. The data are compared to the estimates of the strength of phonon scattering by cation disorder. The thermal conductivity is in good agreement with the theoretical modeling results based on the mass difference for In-rich (x > 0.9) and Ga-rich (x 0.2) compositions. At intermediate compositions (0.2 x 0.9), the thermal conductivity is strongly suppressed below the values expected for homogeneous alloys. We attribute this suppression of thermal conductivity to phonon scattering by nanometer-scale compositional inhomogeneities in alloys.