Summary: This paper deals with the one-machine dynamic total completion time scheduling problem. This problem is known to be NP-hard in the strong sense. A polynomial time heuristic algorithm is proposed which applies the recently introduced Recovering Beam Search (RBS) approach. The algorithm is based on a beam search procedure with unitary beam width and includes a recovering subroutine that allows to overcome wrong decisions taken at higher levels of the beam search tree. It is shown that the total number of considered nodes is bounded by n where n is the jobsize. The proposed algorithm is able to solve in very short CPU time problems with up to 500 jobs outperforming the best state of the art heuristics.