We aim to study the effect of environment on the presence and fuelling of active galactic nuclei (AGNs) in massive galaxy clusters. We explore the use of different AGN detection techniques with the goal of selecting AGN across a broad range of luminosities, AGN/host galaxy flux ratios and obscuration levels. From a sample of 12 galaxy clusters at redshifts 0.5 < z < 0.9, we identify AGN candidates using optical variability from multi-epoch Hubble Space Telescope imaging, X-ray point sources in Chandra images and mid-infrared (IR) spectral energy distribution power-law fits through the Spitzer Infrared Array Camera channels. We find 178 optical variables, 74 X-ray point sources and 64 IR power-law sources, resulting in an average of ∼25 AGN per cluster. We find no significant difference between the fraction of AGN among galaxies in clusters and the percentage of similarly detected AGN in field galaxy studies (∼2.5 per cent). This result provides evidence that galaxies are still able to fuel accretion on to their supermassive black holes, even in dense environments. We also investigate correlations between the percentage of AGN and cluster physical properties such as mass, X-ray luminosity, size, morphology class and redshift. We find no significant correlations among cluster properties and the percentage of AGN detected.