AI systems for mental health are developed predominantly using data from Western, Educated, Industrialized, Rich, and Democratic (WEIRD) populations, raising concerns about their validity, fairness, and generalizability across geo-cultural contexts. This limitation is especially consequential in mental health, where linguistic expression, symptom presentation, help-seeking behavior, and access to care vary substantially across populations. We argue that culturally responsive AI mental health systems require explicit attention to culture throughout the development lifecycle, from data collection to training and deployment. We present a sociotechnical framework for developing culturally responsive AI mental health applications to provide AI researchers and practitioners with an actionable roadmap for building more equitable, reliable, and contextually appropriate mental health technologies.