With the emergence of wireless sensor networks, an enabling communication technology for distributed real-time systems, we face the critical challenge of meeting the end-to-end deadlines of real-time flows. This paper presents Real-time Flow Scheduling (RFS), a novel conflict-free real-time transmission scheduling approach for periodic real-time flows in wireless sensor networks. In contrast to existing transmission scheduling algorithms that ignore interference between transmissions or prevent spatial reuse within the same channel, RFS supports spatial reuse through a novel interference-aware transmission scheduling. While recent work on conflict-free transmission scheduling focused on specialized communication patterns such as queries and converge cast, RFS is designed for peer-to-peer real-time flows with arbitrary inter-flow interference. Moreover, RFS has three salient that make it particularly suitable for real-time systems: First, RFS includes a real-time schedulability analysis that accounts for interference between real-time flows. Second, RFS improves reliability by incorporating retransmissions in a flexible scheduling scheme. Finally, RFS enhances scalability by dividing the network into neighborhoods and provides real-time performance for flows crossing multiple neighborhoods through a novel application of the Release Guard protocol. RFS was evaluated through simulations based on the traces collected from an indoor wireless sensor network test bed. Compared to a traditional TDMA protocol, RFS reduces flow latencies by up to 2.5 times, while improving the real-time capacity by as much as 3.9 times.