Generative models of 3D cardiovascular anatomy can synthesize informative structures for clinical research and medical device evaluation, but face a trade-off between geometric controllability and realism. We propose CardioComposer: a programmable, inference-time framework for generating multi-class anatomical label maps from interpretable ellipsoidal primitives. These primitives represent geometric attributes such as the size, shape, and position of discrete substructures. We specifically develop differentiable measurement functions based on voxel-wise geometric moments, enabling loss-based gradient guidance during diffusion model sampling. We demonstrate that these losses can constrain individual geometric attributes in a disentangled manner and provide compositional control over multiple substructures. Finally, we show that our method is compatible with a broad range of anatomical systems containing non-convex substructures, spanning cardiac, vascular, and skeletal organs. We release our code at https://github.com/kkadry/CardioComposer.