We fabricate a free-standing molybdenum diselenide (MoSe2) saturable absorber by embedding liquid-phase exfoliated few-layer MoSe2 flakes into a polymer film. The MoSe2-polymer composite is used to Q-switch fiber lasers based on ytterbium (Yb), erbium (Er) and thulium (Tm) gain fiber, producing trains of microsecond-duration pulses with kilohertz repetition rates at 1060 nm, 1566 nm and 1924 nm, respectively. Such operating wavelengths correspond to sub-bandgap saturable absorption in MoSe2, which is explained in the context of edge-states, building upon studies of other semiconducting transition metal dichalcogenide (TMD)-based saturable absorbers. Our work adds few-layer MoSe2 to the growing catalog of TMDs with remarkable optical properties, which offer new opportunities for photonic devices.