We present a reduced-complexity approximate density evolution scheme that is particularly suitable for Markov-modulated channels, based on the semi-Gaussian approximation from [1]. We propose a design algorithm whose complexity is comparable to the memoryless case, assuming unlimited precomputation is allowed. We also present degree sequences representing some of the best known codes in the GE channel which were designed using this technique. This design tool can be easily extended to more complicated Markov-modulated channels