A continuous-time finite-state Markov chain observed in white noise is considered. The well-known result of Wonham filter provides a formula for obtaining posterior probabilities. Although the filter is of finite dimension, numerical schemes are needed in applications because of the nonlinearity and because the observations are frequently collected in discrete moments. In this work, we develop approximation schemes ofWonham filters by constructing discrete-time recursive algorithms. We prove the convergence of the algorithm by weak convergence method and martingale averaging techniques. Numerical experiments are also fished to demonstrate the perfomance of our algorithms.