This paper investigates the viscoelastic behavior of polyurethane foams. A three-dimensional single hereditary integral nonlinear viscoelastic model is developed using the results of stress relaxation experiments in compression, tension, and shear. The material behavior is characterized with the instantaneous dilatational and deviatoric elastic responses together with the normalized relaxation functions. The relaxation of the material is described using three different functions for high compression (above 7% strain), low compression/tension, and shear. The material model is implemented in the LS-DYNA3D program and is validated by comparing the numerical and experimental results.