Previously almost all biomechanically-based time-critical surgical simulation has ignored the well established features of tissue mechanical response of anisotropy and time-dependence. We address this issue by presenting an efficient solution procedure for anisotropic viscohyperelastic constitutive models which allows use of these in nonlinear explicit dynamic finite element algorithms. We show that the procedure allows incorporation of both anisotropy and viscoelasticity for as little as 5.1% additional cost compared with the usual isotropic elastic models. When combined with high performance GPU execution the complete framework is suitable for time-critical simulation applications such as interactive surgical simulation and intraoperative image registration.