For R a Galois ring and m 1, . . . , m l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ is an R[x]-submodule of $${R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}$$. Suppose m 1, . . . , m l are all coprime to the characteristic of R and let {g 1, . . . , g t } be the set of all monic basic irreducible polynomials in the factorizations of $${x^{m_i}-1}$$ (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ are identified with $${{\mathcal G}_1\times\cdots\times {\mathcal G}_t}$$, where $${{\mathcal G}_j}$$ is an R[x]/(g j )-submodule of $${(R[x]/(g_j))^{n_j}}$$, where n j is the number of i for which g j appears in the factorization of $${x^{m_i}-1}$$ into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.