The discovery of periodicity in the arrival times of the fast radio bursts (FRBs) poses a challenge to the oft-studied magnetar scenarios. However, models that postulate that FRBs result from magnetized shocks or magnetic reconnection in a relativistic outflow are not specific to magnetar engines; instead, they require only the impulsive injection of relativistic energy into a dense magnetized medium. Motivated thus, we outline a new scenario in which FRBs are powered by short-lived relativistic outflows (``flares) from accreting black holes or neutron stars, which propagate into the cavity of the pre-existing (``quiescent) jet. In order to reproduce FRB luminosities and rates, we are driven to consider binaries of stellar-mass compact objects undergoing super-Eddington mass-transfer, similar to ultraluminous X-ray (ULX) sources. Indeed, the host galaxies of FRBs, and their spatial offsets within their hosts, show broad similarities with ULXs. Periodicity on timescales of days to years could be attributed to precession (e.g., Lens-Thirring) of the polar accretion funnel, along which the FRB emission is geometrically and relativistically beamed, which sweeps across the observer line of sight. Accounting for the most luminous FRBs via accretion power may require a population of binaries undergoing brief-lived phases of unstable (dynamical-timescale) mass-transfer. This will lead to secular evolution in the properties of some repeating FRBs on timescales of months to years, followed by a transient optical/IR counterpart akin to a luminous red nova, or a more luminous accretion-powered optical/X-ray transient. We encourage targeted FRB searches of known ULX sources.
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