A moment method is used to compute the anisotropic spin diffusion constant in two-dimensional (2D) adsorbed and bulk (3D) quantum solids in which the spin motion is induced by an exchange Hamiltonian. Computations are carried out in 2D for the square and triangular lattices and in 3D for the hcp lattice. It is assumed that there are pair and three-particle exchange processes only. Since, in hcp3He, exchange processes out of the basal planeJprime may occur at a different rate from processes in the planeJ, comparison with experimental results on single crystals should allow the determination ofJ andJprime. Our results are given as functions of the ratioy=Jprime/J and of the angle betweenc axis and field gradient. The 2D triangular lattice is shown to correspond to the special casey=0 of the hcp lattice. Our square-lattice result compares well with that of Morita (who used a different technique), supporting the validity of our method.