The economy and flexibility of the pi-calculus make it an attractive object of theoretical study and a clean basis for concurrent language design and implementation. However, such generality has a cost: encoding higher-level features like functional computation in pi-calculus throws away potentially useful information. We show how a linear type system can be used to recover important static information about a process's behaviour. In particular, we can guarantee that two processes communicating over a linear channel cannot interfere with other communicating processes. This enables more aggressive optimisation of communications over linear channels and allows useful refinements to the usual notions of process equivalence for pi-calculus. After developing standard results such as soundness of typing, we focus on equivalences, adapting the standard notion of barbed bisimulation to the linear setting and showing how reductions on linear channels induce a useful ``partial confluence'' of process behaviors.