In order to understand the (polymer-induced) drag-reduction mechanism in wall turbulence of viscoelastic fluid, it must be crucial to elucidate how the viscoelasticity modulates the streamwiseelongated vortical structure, which is a characteristic feature in the near-wall turbulence. In the present study we numerically study the roll-cell structure of laminar rotating plane Couette flow of viscoelastic fluid, aiming at further understanding the effect of viscoelasticity on the longitudinal vortices in shear flow. The case of Re = 25 and Ro = 0:4 is focused as a typical state that provides twodimensional steady roll cells for the Newtonian fluid case, and we investigate how such two-dimensional vortices would be modulated by the viscoelasticity at different Weissenberg numbers. The viscoelasticity is found to give rise to an unsteady flow state where the two-dimensional roll-cell structure is periodically strengthened and suppressed with the time scale of the relaxation time of the fluid viscoelasticity. Such a pulsatile fluid motion was caused by the delay in the response of the viscoelastic force to the change in the velocity gradient, and the effect of the viscoelasticity to the momentum transport in the flow field is also discussed in detailed.