In this paper, a general class of viscoelastic model is used to investigate numerically the pattern and strength of the secondary flows in rectangular pipes as well as the influence of material parameters on them. To solve the coupled governing equation system, an implicit finite volume method based on the SIMPLEST algorithm, which is applicable for both time-dependent and steady-state flow computations, has been developed and extended for viscoelastic flow computations by applying the decoupled techniques. The main feature of the method is to split the solution process into a series of steps in which the continuity of the flow field is enforced by solving a Poisson＇s equation for the pressure, and at the end of the steps, both the pressure and velocity fields are made to satisfy one and the same momentum equation. For viscoelastic flow computations, artificial diffusion terms are introduced on both sides of the discretized constitutive equations to improve numerical stability. It is found that there are in total two vortices in each quadrant of the pipe at different aspect ratios (from 1 to 16), and at each ratio the pattern of secondary flows takes the same form for different material parameters, but their strength is very sensitive to the viscoelastic material parameters. Numerical results indicate that the presence of secondary flow strongly depends on the primary flow rate and the elasticity of the fluid, namely, the first and the second normal stress differences as well as their functional departure from the constant multiple viscosity.