The dramatic increase of mobile devices and mobile subscriptions for delay-sensitive high-rate multimedia services has caused rapidly rising energy consumption, which is a big problem for the next generation wireless networks. It is urgent to find solutions that can satisfy the high data rate quality of service (QoS) requirements from users and obtain high energy-efficiency (EE) for the network at the same time. In this paper, energy-efficient communication is investigated for the orthogonal frequency-division multiple-access (OFDMA) based user-relay aided cellular networks that is envisioned as a promising technology to unlock the full potential of 5G networks. The EE maximization problem is formulated for dual-hop user-relay aided downlink cellular networks. Decode-and-forward (DF) relaying strategy is used by considering the QoS constraints. Since the formulated problem is a mixed-integer non-linear programming problem (MINLP) that is difficult to solve for multi-carrier, multi-user, multi-relay networks, a two-stage radio resource management solution is proposed to tackle the problem. The advantages of the proposed scheme, which not only considers the EE but also meets the data rate requirements of the users, are revealed by performing extensive simulations.