This thesis develops a method for estimating the parameters of continuous-time Markov chains discretely observed by Poisson sampling. The inference problem in this context is usually simplified by assuming the process to be time-homogeneous and that the process can be observed continuously for some observation period. But many real problems are not homogeneous; moreover, in practice it is often difficult to observe random processes continuously. In this work, the Dynkin Identity motivates a martingale estimating equation which is no more complicated a function of the parameters than the infinitesimal generator of the chain. The time-dependent generators of inhomogeneous chains therefore present no new obstacles. The Dynkin Martingale estimating equation derived here applies to processes discretely observed according to an independent Poisson process. Random observation of this kind alleviates the so-called aliasing problem, which can arise when continuous-time processes are observed discretely. Theoretical arguments exploit the martingale structure to obtain conditions ensuring strong consistency and asymptotic normality of the estimators. Simulation studies of a single-server Markov queue with sinusoidal arrivals test the performance of the estimators under different sampling schemes and against the benchmark maximum likelihood estimators based on continuous observation.