In 1999, Bertram, Ciocan-Fontanine and Fulton related quantum multiplication of Schur polynomials to the classical product via rim-hook removal. This is called the "rim-hook rule." Since the Littlewood-Richardson rule is easily accessible, this means that products in QH^*(Gr(k,n)) are also similarly accessible. We provide an equivariant version of this rim-hook rule, explicitly relating the rings QH^*_T(Gr(k,n)) and H^*_T(Gr(k,2n-1)) or alternately the quantum product of factorial Schur polynomials to the classical product. This allows computations in QH^*_T(Gr(k,n)) using combinatorial devices such as Knutson and Tao´s puzzles for H^*_T(Gr(k,n)). Interestingly, this rule requires a specialization of torus weights and a definition of "cyclic factorial Schur functions" that is tantalizingly similar to maps in affine Schubert calculus, which is related to Gromov-Witten theory by Peterson´s theorem.