The quantum Fourier transform (QFT) is the principal ingredient of most efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by ``quantizing'' the highly successful separation of variables technique for the construction of efficient classical Fourier transforms. Specifically, we use Bratteli diagrams, Gel'fand-Tsetlin bases, and strong generating sets of small adapted diameter to provide efficient quantum circuits for the QFT over a wide variety of finite Abelian and non-Abelian groups, including all group families for which efficient QFTs are currently known and many new group families. Moreover, our method provides the first subexponential-size quantum circuits for the QFT over the linear groups GL_k(q), SL_k(q), and the finite groups of Lie type, for any fixed prime power q.