The authors generalize a technique to construct resolvable designs as given by \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson} [Survey Combin. Theory, Sympos. Colorado State Univ., Colorado 1971, 361-375 (1973; Zbl 0274.05010)] using free difference families in finite fields. The generalized techniques require free difference families over rings in which there are some units such that their differences are still units. The new techniques are used to construct resolvable designs, frames, and resolvable (modified) group divisible designs with index not less than unity. The construction technique developed is then applied to construct resolvable perfect Mendelsohn designs and other related designs. Furthermore, the results of this paper are used to obtain some bounds on the largest cardinality of subsets of \(U(R)\), the set of the units of a ring \(R\), in which the differences among elements are still units. Finally, composition theorems for free difference families via difference matrices are stated which can be used to construct new resolvable designs.