Recently, free space optical sensor networks (FSOSNs), which are based on free space optics (FSO) instead of radio frequency (RF), have gained increasing visibility over traditional wireless sensor networks (WSNs) due to their advantages such as larger capacity, higher security, and lower cost. However, the performance of FSOSNs is restricted to the requirement of a direct line-of-sight (LOS) path between a sender and a receiver pair. Once a node dies of energy depletion, the network would probably suffer from a dramatic decrease of connectivity, resulting in a huge loss of data packets. Thus, this paper proposes a reconfigurable routing protocol (RRP) to overcome this problem by dynamically reconfiguring the network virtual topology. The RRP works in three phases: (1) virtual topology construction, (2) routing establishment, and (3) reconfigurable routing. When data transmission begins, the data packets are first routed through the shortest hop paths. Then a reconfiguration is initiated by the node whose residual energy falls below a threshold. Nodes affected by this dying node are classified into two types, namely maintenance nodes and adjustment nodes, and they are reconfigured according to the types. An energy model is designed to evaluate the performance of RRP through OPNET simulation. Our simulation results indicate that the RRP achieves better performance compared with the simple-link protocol and a direct reconfiguration scheme in terms of connectivity, network lifetime, packet delivery ratio and the number of living nodes.