The reliability of consecutive-type systems has been approached from different angles. We present a new method for deriving the generating functions and reliabilities of various consecutive-type systems. Our method, which is based on Feller's run theory, is easy to implement, and leads to both recursive and non-recursive formulas for the reliability. The non-recursive expression is especially advantageous for systems with numerous components. We show how the method can be extended for computing generating functions and reliabilities of systems with multi-state components as well as systems with statistically dependent components. To make our theoretical derivations practical to practitioners, we include short computer programs that do the non-recursive computations yielding the reliabilities of such systems.