We describe an iterative refinement method that is designed to smooth, but not over-smooth, noisy triangle meshes. Our method not only preserves sharp features but also retains visually meaningful fine scale components or features, referred to as 'intrinsic texture'. An anisotropic diffusion method is first developed that is comparable in efficiency and performance to bilateral filtering. This then becomes the inner iteration in an iterative refinement process that gradually and adaptively increases the importance of data fidelity, yielding an efficient multiscale algorithm that is capable of handling irregular meshes and intrinsic texture.