Generative modeling of crystalline materials using diffusion models presents aseries of challenges: the data distribution is characterized by inherent symmetriesand involves multiple modalities, with some defined on specific manifolds. Notably,the treatment of fractional coordinates representing atomic positions in the unitcell requires careful consideration, as they lie on a hypertorus. In this work, weintroduce Kinetic Langevin Diffusion for Materials (KLDM), a novel diffusionmodel for crystalline materials generation, where the key innovation resides in themodeling of the coordinates. Instead of resorting to Riemannian diffusion on thehypertorus directly, we generalize Trivialized Diffusion Models (TDM) to accountfor the symmetries inherent to crystals. By coupling coordinates with auxiliaryEuclidean variables representing velocities, the diffusion process is now offset to aflat space. This allows us to effectively perform diffusion on the hypertorus whileproviding a training objective consistent with the periodic translation symmetry ofthe true data distribution. We evaluate KLDM on both Crystal Structure Predic-tion (CSP) and De-novo Generation (DNG) tasks, demonstrating its competitiveperformance with current state-of-the-art models.