In order to propagate transactions and blocks, today's blockchain systems rely on unstructured peer-to-peer overlay networks. In such networks, broadcast is known to be an inefficient operation in terms of message complexity and overhead. In addition to the impact on the system performance, inefficient or delayed block propagation may have severe consequences regarding security and fairness of the consensus layer. In contrast, the Kadcast protocol is a structured peer-to-peer protocol for block and transaction propagation in blockchain networks. Kadcast utilizes the well-known overlay topology of Kademlia to realize an efficient broadcast operation with tunable overhead. We study the security and privacy of the Kadcast protocol based on probabilistic models and analyze its resilience to packet losses and node failures. Moreover, we evaluate Kadcast's block delivery performance, broadcast reliability, efficiency, and security based on advanced network simulations. Lastly, we introduce a QUIC-based prototype implementation of the Kadcast protocol and show its merits through deployment in a global-scale cloud-based testbed.