We present a robust scheme by which fractional quantum Hall states of bosons can be achieved for ultracold atomic gases. We describe a new form of optical flux lattice, suitable for commonly used atomic species with ground state angular momentum Jg=1, for which the lowest energy band is topological and nearly dispersionless. Through exact diagonalization studies, we show that, even for moderate interactions, the many-body ground states consist of bosonic fractional quantum Hall states, including the Laughlin state and the Moore-Read (Pfaffian) state. These phases are shown to have energy gaps that are larger than temperature scales achievable in ultracold gases.