The general moment-method calculation of diffusion is reviewed for the high-temperature long-wavelength limit. The temporal Fourier transform of the correlation function is assumed Lorentzian with a Gaussian cutoff. Exact expressions for the second and fourth moments are obtained as lattice sums over the spin-spin interactions. For a spin system with H=12i,jAijIiIj, and Aij nonzero only for i, j nearest neighbors, one has D=cAijb2, where b is the nearest-neighbor distance and c=0.328 for the bcc lattice and 0.296 for the sc lattice; the latter agrees with a similar calculation by Mori and Kawasaki. For exchange-energy diffusion, c=0.67 for the bcc lattice. In a classical dipolar spin system, the exact moment method gives nearly the same result as the density matrix theory of Lowe and Gade. Reduction of diffusion by moderate quadrupolar interaction and diffusion energetics in an inhomogeneous field are discussed. © 1968 The American Physical Society.