Brownian dynamics simulations of shear flows were carried out for the suspensions of ellipsoids interacting via the Gay-Berne potential. When the number density of ellipsoid was increased through a compression process of the primitive cell, the phase transition from an isotropic phase to a nematic one occurred. In the simple shear flow for the suspension which exhibits the LC phase at equilibrium, the orientation behavior was examined. The kayaking motion and the intermittent rotation of director were observed at low shear rates. In their director motions, the director is out of the vorticity plane. At high shear rates, however, the flow-aligning was found.