The immersed boundary method is both a general mathematical framework and a particular numerical approach to problems of fluid-structure interaction. In this paper, we describe the application of the immersed boundary method to the simulation of cardiovascular fluid dynamics, focusing on the fluid dynamics of the aortic heart valve (the valve which prevents the backflow of blood from the aorta into the left ventricle of the heart) and aortic root (the initial portion of the aorta, which attaches to the heart). The aortic valve and root are modeled as a system of elastic fibers, and the blood is modeled as a viscous incompressible fluid. Three-dimensional simulation results obtained using a parallel and adaptive version of the immersed boundary method are presented. These results demonstrate that it is feasible to perform three-dimensional immersed boundary simulations of cardiovascular fluid dynamics in which realistic cardiac output is obtained at realistic pressures.