Sufficient conditions are given for effective model-checking of parameterized ring networks of isomorphic finite-state processes. Unlike others appearing in the literature, the present sufficient conditions do not restrict the mechanism whereby processes interact with one another, but rather the structure of the processes themselves. The results provide "cutoffs" for systems of "piecewise recognizable" processes, and show that all ring networks based on a given piecewise recognizable template fall into a finite number of weak trace equivalence classes. This result is then extended to three other finer equivalence relations: complete trace equivalence, weak failure equivalence and weak possible-futures equivalence. The paper also formalizes a notion of processes whose actions affect only a bounded number of other processes, using the property of "shuffled processes"; if a ring segment is a shuffled process, then all ring networks fall into a finite number of "weak bisimilarity" classes. It is also shown that each of the above equivalence relations preserve a subset of "observable modal logic" formulas.