Ethical oversight of AI research is beset by a number of problems. There are numerous ways to tackle these problems, however, they leave full responsibility for ethical reflection in the hands of review boards and committees. In this paper, we propose an alternative solution: the training of ethically responsible AI researchers. We showcase this solution through a case study of a centre for doctoral training and outline how ethics training is structured in the program. We go on to present two second-year students' reflections on their training which demonstrates some of their newly found capabilities as ethically responsible researchers.