The expanding jets and lobes of radio AGN are associated with one of the most important forms of feedback, directly affecting star formation histories of host galaxies and influencing the shape of the galaxy mass function. Understanding how they are triggered is key to properly incorporate radio AGN into models of galaxy evolution. They can be characterised by their accretion modes, with the radiatively-efficient radio AGN (e.g. HERGs) accreting at a high Eddington rate (>1%), in contrast with their radiatively-inefficient counterparts (e.g. LERGs) at much lower Eddington rates. Previous studies have suggested that the dominant triggering mechanisms vary between these subtypes, but at low levels of statistical significance. We present the results from a deep INT/WFC imaging survey of a complete sample of all 112 3CR radio galaxies at z < 0.3, along with a stellar mass and redshift matched control sample. The results provide the first clear evidence for significant differences (~3s) between the triggering mechanisms of the different subtypes of the powerful radio AGN population. The HERGs show a high rate of morphological disturbance (~ 4s differences with the control sample), consistent with triggering in galaxy mergers and interactions. The LERGs show a much lower rate of morphological disturbance, consistent with the control sample, suggesting that they are triggered via the accretion of gas from the hot X-ray haloes of galaxy clusters.