The objective of the paper is to introduce a novel energy-aware scheduling (EAS) algorithm which statically schedules application-specific communication transactions and computation tasks onto heterogeneous network-on-chip (NoC) architectures. The proposed algorithm automatically assigns the application tasks onto different processing elements and then schedules their execution under real-time constraints. At the same time, the algorithm takes into consideration the exact communication delay by scheduling communication transactions in parallel. As the main theoretical contribution, the authors first formulate the problem of concurrent communication and task scheduling for heterogeneous NoC architectures and then propose an efficient heuristic to solve it. Experimental results show that significant energy savings can be achieved while meeting the specified performance constraints. For instance, for a complex multimedia application, 31% energy savings have been observed, on average, compared to the schedules generated by a standard earliest-deadline-first scheduler.