Internal rotation is thought to play a major role in the dynamics of some globular clusters. However, in only a few cases has internal rotation been studied by the quantitative application of realistic and physically justified global models. Here, we present a dynamical analysis of the photometry and three-dimensional kinematics of ω Cen, 47 Tuc, and M15, by means of a recently introduced family of self-consistent axisymmetric rotating models. The three clusters, characterized by different relaxation conditions, show evidence of differential rotation and deviations from sphericity. The combination of line-of-sight velocities and proper motions allows us to determine their internal dynamics, predict their morphology, and estimate their dynamical distance. The well-relaxed cluster 47 Tuc is interpreted very well by our model; internal rotation is found to explain the observed morphology. For M15, we provide a global model in good agreement with the data, including the central behavior of the rotation profile and the shape of the ellipticity profile. For the partially relaxed cluster ω Cen, the selected model reproduces the complex three-dimensional kinematics; in particular, the observed anisotropy profile, characterized by a transition from isotropy to weakly radial anisotropy and then to tangential anisotropy in the outer parts. The discrepancy found for the steep central gradient in the observed line-of-sight velocity dispersion profile and for the ellipticity profile is ascribed to the condition of only partial relaxation of this cluster and the interplay between rotation and radial anisotropy.