Constructing Markov chains with desired statistical properties is of critical importance for many applications in economics and finance. This paper proposes a moment-matching method for generating finite-state Markov chains with flexible distributions, including empirically-relevant processes with non-zero skewness and excess kurtosis. In our method, we preserve the most appealing features of the existing methods, including full analytical tractability and minimal computational cost. The new method derives the key moments of the Markov chain as closed-form functions of its parameters. It thus offers a simple plug-in procedure requiring neither numerical integration nor optimization. Using the method amounts to plugging the targeted values of mean, standard deviation, serial correlation, skewness, and excess kurtosis into a simple procedure. The proposed method outperforms the existing methods over a wide range of the parameter space, especially for leptokurtic processes with characteristic roots close to unity. The supplementary materials include ready-to-use computer codes of the plug-in procedures and practical guidelines.