Intelligent reflecting surface (IRS) is an emerging technology to enhance the spectral and energy efficiency of wireless communications cost-effectively. This letter considers a new multi-IRS aided wireless network where a cascaded line-of-sight (LoS) link is established between the base station (BS) and a remote user by leveraging the multi-hop signal reflection of selected IRSs. As compared to the conventional single-/double-hop IRS system, multi-hop IRS system provides more pronounced path diversity and cooperative passive beamforming gains, especially in the environment with dense obstacles. However, a more challenging joint active/passive beamforming and multi-hop beam routing problem also arises for maximizing the end-to-end channel gain. Furthermore, the number of IRS-associated channel coefficients increases drastically with the number of IRS hops. To tackle the above issues, in this letter we propose a new and efficient beam training based solution by considering the use of practical codebook-based BS/IRS active/passive beamforming without the need of explicit channel estimation. Instead of exhaustively or sequentially searching over all combinations of active and passive beam patterns for each beam route, a distributed beam training scheme is proposed to reduce the complexity, by exploiting the (nearly) time-invariant BS-IRS and inter-IRS channels and the cooperative training among the BS and IRSs' controllers. Simulation results show that our proposed design achieves the end-to-end channel gain close to that of the sequential beam search, but at a much lower training overhead and complexity.