I propose a new radiation-generating device called the laser undulator x-ray source (LUXUS). In particular, I demonstrate that LUXUS has the potential to generate a highly-directional, monochromatic beam of x-rays which is continuously tunable over multiple orders of magnitude in photon energy. In the configuration used as a model, the photons will be seen to have energies lying between 18 eV and 45 KeV when the electron accelerator energy is varied over the range 1 - 50 MeV. Reduced to its essentials, LUXUS consists of (1) a high-current, cyclic electron induction accelerator (such as a 'rebatron') in a racetrack configuration, (2) a high-finesse Fabry-Perot resonator used as a power-buildup cavity, and (3) a high-power, CW, injection-locked Nd:YAG laser to which, in turn, the Fabry-Perot resonator is locked. With a mean accelerator current of 1000 A and a Nd:YAG laser of 15 W output, an x-ray flux approximately 1013 photon/s should be possible at any selected energy within the x-ray range mentioned above (18 eV to 45 KeV). Important applications in medicine--including digital subtractive angiography, x-ray mammography, and general monochromatic radiography--will be discussed briefly.