We present a novel approach for simulating blood flow dynamics in cardiovascular modeling using a modified version of MeshGraphNet, a graph neural network architecture originally developed for meshed data. Our method involves developing one-dimensional reduced-order models that predict the pressure and flow rate at vessel centerline nodes. The graph-neural network acts as an iterative solver, taking the state of the system at a particular timestep as input and providing an update that allows us to evolve the system to the next timestep. The approach is accurate and generalizable, achieving errors below 2% and 3% for pressure and flow rate, respectively, in a variety of anatomies and boundary conditions. Our modifications to MeshGraphNet enable its application to our specific problem domain, and our findings demonstrate the effectiveness of our approach for simulating blood flow dynamics in complex cardiovascular systems.