We report, for the first time to our knowledge, the experimental demonstration of multi-cavity optoelectronic oscillators where the cavities are provided by the different cores of a multicore fiber. We implemented two multi-cavity architectures over a 20-m-long 7-core fiber link: unbalanced dual-cavity oscillation (the cavity lengths are a multiple of a reference value) and multi-cavity Vernier oscillation (the cavity lengths are slightly different). Since all the cavities are hosted under a single fiber cladding and are subject to the same environmental and mechanical conditions, optoelectronic oscillators built upon multicore fibers benefit from improved performance stability as compared to independent singlemode fiber cavities.