The theory of self-ordering of multisheet arrays of two-dimensional islands is developed where the structure of the surface sheet is determined by thermodynamic equilibrium in the strain field created by a fixed structure of the sheets of buried islands. It is shown that, in cubic crystals with a negative parameter of elastic anisotropy, xi = (c(11) - c(12) -2c(44))/c(44) < 0, the elastic interaction between successive sheets of islands parallel to the (001) crystallographic plane exhibits an oscillatory decay with the separation between sheets. This oscillatory decay is related to generalized Rayleigh waves in elastically anisotropic crystals. By varying the distance between successive sheets of islands, a transition occurs from vertical correlations between islands where islands of the upper sheet are formed above the buried islands of the lower sheet to anti-correlations between islands where islands of the upper sheet are formed above the spacings in the lower sheet. Thus an explanation for the recently observed anti-correlations in multisheet arrays of CdSe islands in ZnSe matrix is obtained.