The equations for viscoelastic flows of an Oldroyd-B fluid are integrated using the finite volume technique. The numerical algorithm was developed to treat three-dimensional (3D) unsteady flows using Cartesian coordinates on a non-uniform staggered grid. The primitive variables, velocities, pressure and extra-stresses are used in the formulation. All inertia terms in the momentum and constitutive equations are taken into account and are discretized in space using a quadratic upwind scheme. Case studies have been conducted for the start-up Couette flow, two-dimensional (2D) 4:1 and 3D 4:1:4 planar contractions. The numerical solutions agree very well with analytical solutions for the start-up Couette flow. The size of the corner vortex for the 4:1 planar contraction, in the 2D case, is in good agreement with previous computations. Comparison between 2D calculation for a qualitative analysis, using the Oldroyd-B fluid, and measurements [1] of a 5.0 wt.% solution of polyisobutylene in tetradecane, is presented for the velocity and normal stress difference at several cross sections in the planar contraction. New results showing the vector field, streamlines and extra-stress components are presented for a 3D 4:1:4 planar contraction at high Deborah numbers (27.3).