The unsteady flow of an incompressible generalized Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate has been studied using Fourier sine and Laplace transforms. The obtained solutions for the velocity field and shear stresses, written in terms of the generalized G and R functions, are presented as sum of the similar Newtonian solutions and the corresponding non-Newtonian contributions. For alpha = beta = 1 and lambda(r) -> lambda these solutions are going to the corresponding Newtonian solutions. Furthermore, the solutions for generalized Maxwell fluids as well as those for ordinary Oldroyd-B and Maxwell fluids, performing the same motion, are also obtained as limiting cases of our general solutions. In the absence of the side walls, namely when the distance between the two walls tends to infinity, the solutions corresponding to the motion over an infinite constantly accelerating plate are recovered. For lambda(r) -> 0 and beta -> 1, these solutions reduce to the known solutions from the literature. Finally, the effect of the material parameters on the velocity profile is spotlighted by means of the graphical illustrations. (C) 2008 Elsevier B.V. All rights reserved.