The change of diffusion kinetics when elastic fields are present is discussed for diffusion on (001) surfaces of simple cubic, fcc and bcc lattices. All particles interact pairwise with a Lennard-Jones potential. The simple cubic lattice was stabilized by an anisotropic prefactor. It is found that generically compressive strain enhances diffusion whereas tensile strain increases the activation barrier. An approximately linear dependence of the barrier in a wide range of misfits is found. In heteroepitaxy, diffusion on top of large clusters is inhomogeneous and anisotropic. The kinetics close to edges and centers of islands are remarkably different. In many cases changes of binding energies are small compared to those of saddle point energies. Thermodynamic arguments (minimization of free energy) are not appropriate to describe diffusion on strained surfaces in these cases.