Planning for autonomous driving in complex, urban scenarios requires accurate prediction of the trajectories of surrounding traffic participants. Their future behavior depends on their route intentions, the road-geometry, traffic rules and mutual interaction, resulting in interdependencies between their trajectories. We present a probabilistic prediction framework based on a dynamic Bayesian network, which represents the state of the complete scene including all agents and respects the aforementioned dependencies. We propose Markovian, contextdependent motion models to define the interaction-aware behavior of drivers. At first, the state of the dynamic Bayesian network is estimated over time by tracking the single agents via sequential Monte Carlo inference. Secondly, we perform a probabilistic forward simulation of the network's estimated belief state to generate the different combinatorial scene developments. This provides the corresponding trajectories for the set of possible, future scenes. Our framework can handle various road layouts and number of traffic participants. We evaluate the approach in online simulations and real-world scenarios. It is shown that our interaction-aware prediction outperforms interaction-unaware physics-and map-based approaches.