Data transmission techniques in wireless relay networks can be improved in terms of energy-efficiency (EE), throughput and complexity. In this paper we propose an energy-efficient distributed space-time block code (DSTBC) technique where the instantaneous channel state information (CSI) is available at all receivers (source and relays) and just the average CSI is valid at the transmitter side. A two-hop protocol is considered where the source sends data to a number of relays which perform decode-and-forward (DF) relaying strategy and a subset of these relays synchronously send data to destination via DSTBC technique. It is assumed that every node in the network has its own power constraint. The effect of optimum rate adaptation for source-to-relay and relay-to-destination links, separately, on EE and throughput is investigated where optimum rate adaptation is considered in addition to transmit power optimization. It is shown that the rate adaptation is very decisive on EE and throughput performances of the network and when it is compared with direct transmission and the case where just transmission power optimization is performed without rate adaptation, the proposed technique significantly outperforms.