We demonstrate an all-sky search for persistent, narrowband gravitational waves using mock data. The search employs radiometry to sidereal-folded data in order to uncover persistent sources of gravitational waves with minimal assumptions about the signal model. The method complements continuous-wave searches, which are finely tuned to search for gravitational waves from rotating neutron stars while providing a means of detecting more exotic sources that might be missed by dedicated continuous-wave techniques. We apply the algorithm to simulated Gaussian noise at the level of LIGO design sensitivity. We project the strain amplitude sensitivity for the algorithm for a LIGO network in the first observing run to be $h_0 approx 1.2 times 10^{-24}$ ($1%$ false alarm probability, $10%$ false dismissal probability). We include treatment of instrumental lines and detector artifacts using time-shifted LIGO data from the first observing run.