Complex simulation of physical systems is an invaluable tool for a large number of fields, including engineering and scientific computing. To overcome the computational requirements of high-accuracy solvers, learned graph neural network simulators have recently been introduced. However, these methods often require a large number of nodes and edges, which can hinder their performance. Moreover, they cannot evaluate continuous solutions in space and time due to their inherently discretized structure. In this paper, we propose GraphSplineNets, a method based on graph neural networks and orthogonal spline collocation (OSC) to accelerate learned simulations of physical systems by interpolating solutions of graph neural networks. First, we employ an encoder-decoder message passing graph neural network to map the location and value of nodes from the physical domain to hidden space and learn to predict future values. Then, to realize fully continuous simulations over the domain without dense sampling of nodes, we post-process predictions with OSC. This strategy allows us to produce a solution at any location in space and time without explicit prior knowledge of underlying differential equations and with a lower computational burden compared to learned graph simulators evaluating more space-time locations. We evaluate the performance of our approach in heat equation, dam breaking, and flag simulations with different graph neural network baselines. Our method shows is consistently Pareto efficient in terms of simulation accuracy and inference time, i.e. 3x speedup with 10% less error on flag simulation.