We relate the diagonal and off-diagonal elements of the effective self-- diffusion tensor, D, to the echo intensity in NMR spin-echo experiments. This relationship is used to design pulse sequences from which D can be estimated. This estimate is validated using both isotropic and anisotropic media. One potential application of this work is to use D to determine fiber orientation in vivo. Specifically, the components of D are used to infer fiber orientation, mean diffusion distances, and tissue microstructure within a voxel. D is used to construct a diffusion ellipsoid in a voxel that depicts the local fiber orientation and mean diffusion distances. The eigenvectors and eigenvalues of D are its principal axes and principal diffusion coefficients. Three scalar invariants of D that are independent of the reference frame in which both magnetic field gradients and D are measured, provide information about tissue microstructure that can be imaged in biological tissue and other anisotropic media.