This technical note extends the feedback particle filter (FPF) methodology and algorithms to the problem of filtering a continuous-time Markov chain. The main contribution is the development of a feedback control-based transformation of the Wonham filter, where the control input is realized via a time-modulated Poisson counter process. A complete characterization of the feedback mechanism that defines the FPF is obtained, which leads to tractable algorithms for the nonlinear filtering problem even for large state spaces. Numerical examples are introduced to help illustrate the application of these techniques.