In this paper, by exploiting the special features of temporal correlations of dynamic sparse channels that path delays change slowly over time but path gains evolve faster, we propose the structured matching pursuit (SMP) algorithm to realize the reconstruction of dynamic sparse channels. Specifically, the SMP algorithm divides the path delays of dynamic sparse channels into two different parts to be considered separately, i.e., the common channel taps and the dynamic channel taps. Based on this separation, the proposed SMP algorithm simultaneously detects the common channel taps of dynamic sparse channels in all time slots at first, and then tracks the dynamic channel taps in each single time slot individually. Theoretical analysis of the proposed SMP algorithm provides a guarantee that the common channel taps can be successfully detected with a high probability, and the reconstruction distortion of dynamic sparse channels is linearly upper bounded by the noise power. Simulation results demonstrate that the proposed SMP algorithm has excellent reconstruction performance with competitive computational complexity compared with conventional reconstruction algorithms.