For compressive sensing of dynamic sparse signals, we develop an iterative greedy search algorithm based on subspace pursuit (SP) that can incorporate sequential predictions, thereby taking advantage of its low complexity while improving recovery performance by exploiting correlations described by a state space model. The algorithm, which we call dynamic subspace pursuit (DSP), is presented and experimentally validated. It exhibits a graceful degradation at deteriorating signal conditions while capable of yielding substantial performance gains as conditions improve.