In this paper, we propose a novel semi-directional flooding (SDF) algorithm for on-demand routing in IEEE 802.15.4-based low-rate wireless mesh networks (LRWMNs). The novelty of our work is that the proposed routing algorithm enables route discovery request (RREQ) packets to be semi-directionally flooded, with respect to a source-destination pair, in a fully distributed manner without either physical or virtual location information. The idea behind this is to exploit the hierarchical addressing structure that allows each router node to compute logical tree distances between source-destination pairs, which approximately mirrors relative hop distance between the pairs, without message exchanges. By exploiting tree distances with a given target address, the SDF algorithm enables RREQ packets to be flooded over a small area, directed towards the destination. We also apply SDF to a lightweight on-demand routing algorithm like AODVjr, and propose two techniques of adaptive timers and route repair that reduces path setup delays and overheads during route rediscovery, respectively. We carry out extensive simulations and quantitatively show that SDF significantly reduces flooding overhead of network-wide flooding (NWF). We compare SDF-based AODVjr with both NWF-based AODVjr and enhanced hierarchical routing protocol (EHRP) for on demand routing and demonstrate that SDF-based AODVjr drastically reduces route discovery overhead by up to about 95 percent while still providing comparable or better performance than NWF-based AODVjr and EHRP.