In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier-Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field. Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.