We present a Metropolis-Hastings Markov chain Monte Carlo (MCMC) algorithm for detecting hidden variables in a continuous time Bayesian network (CTBN), which uses reversible jumps in the sense defined by (Green 1995). In common with several Monte Carlo algorithms, one of the most recent and important by (Rao and Teh 2013), our algorithm exploits uniformization techniques under which a continuous time Markov process can be represented as a marked Poisson process. We exploit this in a novel way. We show that our MCMC algorithm can be more efficient than those of likelihood weighting type, as in (Nodelman et al. 2003) and (Fan et al. 2010) and that our algorithm broadens the class of important examples that can be treated effectively.