Providing functionality through articulation and interaction with objects is a key objective in 3D generation. We introduce MIDGArD (Modular Interpretable Diffusion over Graphs for Articulated Designs), a novel diffusion-based framework for articulated 3D asset generation. MIDGArD improves over foundational work in the field by enhancing quality, consistency, and controllability in the generation process. This is achieved through MIDGArD's modular approach that separates the problem into two primary components: structure generation and shape generation. The structure generation module of MIDGArD aims at producing coherent articulation features from noisy or incomplete inputs. It acts on the object's structural and kinematic attributes, represented as features of a graph that are being progressively denoised to issue coherent and interpretable articulation solutions. This denoised graph then serves as an advanced conditioning mechanism for the shape generation module, a 3D generative model that populates each link of the articulated structure with consistent 3D meshes. Experiments show the superiority of MIDGArD on the quality, consistency, and interpretability of the generated assets. Importantly, the generated models are fully simulatable, i.e., can be seamlessly integrated into standard physics engines such as MuJoCo, broadening MIDGArD's applicability to fields such as digital content creation, meta realities, and robotics.