Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use previously generated GMI cuts. We demonstrate that for problems in MIPLIB 3.0 and MIPLIB 2003, the cuts we generate form an important subclass of all rank-1 mixed-integer rounding cuts. Further, we use our heuristic to generate globally valid rank-1 GMI cuts at nodes of a branch-and-cut tree and use these cuts to solve a difficult problem from MIPLIB 2003, namely timtab2, without using problem-specific cuts.