Highly sensitive photodetectors for the mid-infrared have recently been obtained by placing a photodiode inside a Fabry- Pérot cavity (Arnold et al 2005 Appl. Phys. Lett. 87 141103). These resonant cavity enhanced detectors (RCED) are sensitive at the resonances which depend on the distance between the two mirrors of the cavity (Unlu and Strite 1995 J. Appl. Phys. 78 608). Displacing one of these mirrors then makes it possible to change the cavity length and thus select the detection wavelength. The design, simulation and fabrication of a MEMS mirror is discussed in detail and its integration with the counter mirror and the photodiode, both grown by molecular beam epitaxy, are presented. Results obtained with external mirrors moved by piezoactuation as well as first measurements of a complete tunable RCED using electrostatically driven vertically moving MEMS mirrors are presented. Such tunable detectors are of interest in thermographic and spectroscopic applications.