In this paper, we provide a novel approach for studying heavy-tailed asymptotics for the stationary probability vector of a Markov chain of GI/G/1 type, whose transition matrix is constructed by two matrix sequences referred to as a boundary matrix sequence and a repeating matrix sequence. We first provide a necessary and sufficient condition under which the stationary probability vector is heavy-tailed. Then we derive long-tailed asymptotics for the R-measure in terms of the RG-factorization for the repeating matrix sequence and a Wiener-Hopf equation for the boundary matrix sequence. Based on this, we are able to provide a detailed analysis for subexponentail asymptotics of the stationary probability vector.