Deadlocks are a bane for network designers, be it a Network on Chip (NoC) in a multi-core or a large scale HPC/datacenter network. A routing deadlock occurs when there is a cyclic dependence between the buffers of network routers. Most modern systems avoid deadlocks by placing routing restrictions or adding extra virtual channels, in turn hurting performance and adding overhead respectively. In this work, we demonstrate that instead of placing such restrictions, we can, in fact, design routers to themselves guarantee deadlock-freedom, by (i) ensuring that every router always has at least one bubble (i.e., free buffer slot) at any input port, and (ii) this bubble pro-actively moves between input ports. We call this a Brownian Bubble Router (BBR). A BBR guarantees forward progress in any network topology, without requiring any routing restrictions or additional virtual channels. With our BBR design we provide 4× better throughput over state of art deadlock recovery schemes and 40% better throughput over traditional deadlock avoidance schemes in a 8×8 Mesh at negligible area and power overheads.