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Realistic scene-level multi-agent motion simulations are crucial for developing and evaluating self-driving algorithms. However, most existing works focus on generating trajectories for a certain single agent type, and typically ignore the consistency of generated trajectories. In this paper, we propose a novel framework based on diffusion models, called SceneDM, to generate joint and consistent future motions of all the agents, including vehicles, bicycles, pedestrians, etc., in a scene. To enhance the consistency of the generated trajectories, we resort to a new Transformer-based network to effectively handle agent-agent interactions in the inverse process of motion diffusion. In consideration of the smoothness of agent trajectories, we further design a simple yet effective consistent diffusion approach, to improve the model in exploiting short-term temporal dependencies. Furthermore, a scene-level scoring function is attached to evaluate the safety and road-adherence of the generated agent's motions and help filter out unrealistic simulations. Finally, SceneDM achieves state-of-the-art results on the Waymo Sim Agents Benchmark. Project webpage is available at https://alperen-hub.github.io/SceneDM.