It is known that greedy methods perform well for maximizing\textitmonotone submodular functions. At the same time, such methods perform poorly in the face of non-monotonicity. In this paper, we show—arguably, surprisingly—that invoking the classical greedy algorithm -times leads to the (currently) fastest deterministic algorithm, called RepeatedGreedy, for maximizing a general submodular function subject to -independent system constraints. RepeatedGreedy achieves approximation using function evaluations (here, and denote the size of the ground set and the maximum size of a feasible solution, respectively). We then show that by a careful sampling procedure, we can run the greedy algorithm only\textitonce and obtain the (currently) fastest randomized algorithm, called SampleGreedy, for maximizing a submodular function subject to -extendible system constraints (a subclass of -independent system constrains). SampleGreedy achieves -approximation with only function evaluations. Finally, we derive an almost matching lower bound, and show that no polynomial time algorithm can have an approximation ratio smaller than . To further support our theoretical results, we compare the performance of RepeatedGreedy and SampleGreedy with prior art in a concrete application (movie recommendation). We consistently observe that while SampleGreedy achieves practically the same utility as the best baseline, it performs at least two orders of magnitude faster.