In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this article we study left-invariant diffusion on the 3D Euclidean motion group SE(3) and its application to crossing-preserving smoothing of high angular resolution diffusion imaging (HARDI), which is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space R3 o S2 of positions and orientations embedded in SE(3) and can be solved by R3 o S2-convolution with the corresponding Green’s functions. We provide analytic approximation formulae and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (non-linear) convection-diffusions on HARDI-data we put emphasis on the underlying differential geometry on SE(3). We write our left-invariant diffusions in covariant derivatives on SE(3) using the Cartan-connection. This Cartan-connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI-data containing crossing-fibers.