In a relay enhanced system the classical HARQ schemes should be adapted to exploit the better channel conditions of the relay in comparison to the direct link. In particular, we need to decide which node (either the source or one of the relays) should retransmit and how many resources have to be spent to avoid further retransmissions. This adaptation should be optimized with respect to the trade-off between Frame Error Rate (FER) and the required resources. We propose an approach based on accumulated Mutual Information (MI) for relay networks applying different relay functions. We consider several memoryless relay functions without channel decoding aiming for simple and cheap relays. Beside the well-known Amplify-Forward (AF) and Detect-Forward (DetF) schemes, we also consider a soft-information relaying function called Estimate-Forward (EF) as this provides higher Mutual Information than AF and DetF. We show that our proposed HARQ retransmission scheme can improve the throughput significantly, especially if EF is applied at the relay.