In this paper, we address Quality-of-Service (QoS)-aware routing issue for Body Sensor Networks (BSNs) in delay and reliability domains. We propose a data-centric multiobjective QoS-Aware routing protocol, called DMQoS, which facilitates the system to achieve customized QoS services for each traffic category differentiated according to the generated data types. It uses modular design architecture wherein different units operate in coordination to provide multiple QoS services. Their operation exploits geographic locations and QoS performance of the neighbor nodes and implements a localized hop-by-hop routing. Moreover, the protocol ensures (almost) a homogeneous energy dissipation rate for all routing nodes in the network through a multiobjective Lexicographic Optimization-based geographic forwarding. We have performed extensive simulations of the proposed protocol, and the results show that DMQoS has significant performance improvements over several state-of-the-art approaches.