We consider Markovian random evolutions performed by a particle mov- ing in R2 and R3 with some finite constant speed v randomly changing its directions at Poisson-paced time instants of intensity λ > 0 uniformly on the S2 and S3-spheres, respectively. We prove that under the Kac condition v → ∞, λ → ∞, v2/λ → c, c > 0 the transition laws of the motions weakly converge in an appropriate Banach space to the transition law of the two- and three-dimensional Wiener process, respectively, with explicitly given generators