A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2ⁿ binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q_n, and a cyclic Gray code as a Hamiltonian cycle of Q_n. In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u,v of Q_n, n≥4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary, we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q_n has a (cyclic) Gray code that avoids M if and only if Q_n-M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q_n can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamiltonicity of Q_n with faulty edges, which is NP-complete in general, becomes polynomial for up to 2^n-1 edges provided they form a matching.