We address the challenge of generating 3D articulated objects in a controllable fashion. Currently, modeling articulated 3D objects is either achieved through laborious manual authoring, or using methods from prior work that are hard to scale and control directly. We leverage the interplay between part shape, connectivity, and motion using a denoising diffusion-based method with attention modules designed to extract correlations between part attributes. Our method takes an object category label and a part connectivity graph as input and generates an object's geometry and motion parameters. The generated objects conform to user-specified constraints on the object category, part shape, and part articulation. Our experiments show that our method outperforms the state-of-the-art in articulated object generation, producing more realistic objects while conforming better to user constraints.