We investigate the dependence of the visible laser performance of Tb3+:LiLuF3 (Tb:LLF) on the ultraviolet (UV) pumping wavelength and present the first, to the best of our knowledge, UV-laser-diode-pumped Tb3+-based laser. We find an onset of thermal effects already at moderate pump power for UV pump wavelengths with strong excited-state absorption (ESA), which vanishes at pump wavelengths with weak ESA. Pumping with a UV laser diode emitting at 378.5 nm enables continuous wave laser operation in a 3-mm short Tb3+(28 at.%):LLF crystal. Slope efficiencies of 36% at 542/544 nm and 17% at 587 nm are obtained with a minimum laser threshold as low as 4 mW.