We propose a simple and novel framework for MCMC inference in continuous-time discrete-state systems with pure jump trajectories. We construct an exact MCMC sampler for such systems by alternately sampling a random discretization of time given a trajectory of the system, and then a new trajectory given the discretization. The first step can be performed efficiently using properties of the Poisson process, while the second step can avail of discrete-time MCMC techniques based on the forward-backward algorithm. We compare our approach to particle MCMC and a uniformization-based sampler, and show its advantages.