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Markov chains with block-structured transition matrices find many applications in various areas. Such Markov chains are characterized by partitioning the state space into subsets called levels, each level consisting of a number of stages. Examples include Markov chains of GI/M/1 type and M/G/1 type, and, more generally, Markov chains of Toeplitz type, or GI/G/1 type. In the analysis of such Markov chains, a number of properties and measures which relate to transitions among levels play a dominant role, while transitions between stages within the same level are less important. The censoring technique has been frequently used in the literature in studying these measures and properties. In this paper, we use this same technique to study block-structured Markov chains. New results and new proofs on factorizations and convergence of algorithms will be provided.