Peer-to-peer overlay technologies offer several advantages over centralized solutions to managing desktop Grids. We present a new hybrid approach to resource discovery in P2P Grids, i.e. desktop Grids based on peer-to-peer overlays. This approach combines the advantages of information propagation based on spanning trees in chord-like structured overlays and epidemic algorithms. We provide a mathematical model for predicting the process of information dissemination and verify its prediction quality in various evaluations. Moreover, we show the failure resistance of the proposed approach in several scenarios. In particular, we demonstrate the efficiency of our approach even in scenarios where 50% of the peers in the overlay fail in short time.