A generalized Gray map for codes over the ring $\F_q[u]/\gen{u^{t+1}}$ is introduced, where q = p^m is a prime power. It is shown that the generalized Gray image of a linear length-N (1-u^t)-cyclic code over $\F_q[u]/\gen{u^{t+1}}$ is a distance-invariant linear length-q^tN quasi-cyclic code of index q^t / p over $\F_q$. It turns out that if (N, p)=1 then every linear code over $\F_q$ that is the generalized Gray image of a length-N cyclic code over $\F_q[u]/\gen{u^{t+1}}$, is also equivalent to a linear length-q^tN quasi-cyclic code of index q^t/p over $\F_q$. The relationship between linear length-pN cyclic codes with (N, p)=1 over $\F_p$ and linear length-N cyclic codes over $\F_p+u\F_p$ is explicitly determined.