A general scheme is presented for obtaining systematic relativistic corrections to quantum-mechanical sum rules for various calculations in atomic physics. The single-particle picture and the Foldy-Wouthuysen transformation [Phys. Rev. 78, 29 (1950)] are adopted to derive such corrections. Results are obtained for the ``generalized Bethe sums,'' which include all higher moments of the energy transfer, and explicit results are given to the lowest order of relativistic corrections.