Opto-mechanical oscillators that generate coherent acoustic waves are drawing much interest, in both fundamental research and applications. Narrowband oscillations can be obtained through the introduction of feedback to the acoustic wave. Most previous realizations of this concept, sometimes referred to as "phonon lasers", relied on radiation pressure and moving boundary effects in micro- or nano-structured media. Demonstrations in bulk crystals required cryogenic temperatures. In this work, stimulated emission of highly-coherent acoustic waves is achieved in a commercially-available multi-core fiber, at room temperature. The fiber is connected within an opto-electronic cavity loop. Pump light in one core is driving acoustic waves via electrostriction, whereas an optical probe wave at a different physical core undergoes photo-elastic modulation by the stimulated acoustic waves. Coupling between pump and probe is based entirely on inter-core, opto-mechanical cross-phase modulation: no direct optical feedback is provided. Single-frequency mechanical oscillations at hundreds of MHz frequencies are obtained, with side-mode suppression that is better than 55 dB. A sharp threshold and rapid collapse of the linewidth above threshold are observed. The linewidths of the acoustic oscillations are on the order of 100 Hz, orders of magnitude narrower than those of the pump and probe light sources. The relative Allan's deviation of the frequency is between 0.1-1 ppm. The frequency may be switched among several values by propagating the pump or probe waves in different cores. The results may be used in sensing, metrology and microwave-photonic information processing applications.