We describe a fill-reducing ordering algorithm for sparse, nonsymmetric LU factorizations, where the pivots are restricted to the diagonal and are selected greedily. The ordering algorithm only uses the structural information. Most of the existing methods are based on some type of symmetrization of the original matrix. Our algorithm exploits the nonsymmetric structure of the given matrix as much as possible. The new algorithm is thus more complex than classical symmetric orderings but we show that our algorithm can be implemented in space bounded by the number of nonzero entries in the original matrix, and has the same time complexity as the analogous algorithms for symmetric matrices. We provide numerical experiments to demonstrate the ordering quality and the runtime of the new ordering algorithm.