In networks that operate during a long time, the routing protocol might have to be changed (in order to apply a routing protocol update, or to take into account a change in the routing metrics). A loop-free transition algorithm has to be used in order to perform the transition to the new routing protocol without generating transient routing loops. In this paper, we propose a loop-free transition algorithm called ACH (avoiding cycles heuristic), which is able to perform the transition in a very small number of steps. Compared to other algorithms of the literature, ACH yields a number of steps which is independent of both the number of nodes and the number of destinations, and thus allows the transition to be performed in a small time. We show through simulations that ACH significantly outperforms other heuristics of the literature, due to its capability to deal with several destinations at once, and due to a priority-based procedure to avoid cycles.