Exact solutions for some oscillating motions of Oldroyd-B fluids, between two infinite coaxial circular cylinders, are established by means of the Laplace transform. These solutions, presented as sum of the steady-state and transient solutions describe the motion of the fluid at small and large times and reduce to the similar solutions for Maxwell, second grade and Newtonian fluids as limiting cases. After some time, when the transients disappear, the starting solutions tend to the steady-state solutions which are periodic in time and independent of the initial conditions. The required time to obtain the steady-state for the cosine and the sine oscillations of the boundary is determined by graphical illustrations. This time decreases if the frequency of the velocity of the boundary increases.