As generative AI technologies are increasingly being launched across the globe, assessing their competence to operate in different cultural contexts is exigently becoming a priority. While recent years have seen numerous and much-needed efforts on cultural benchmarking, these efforts have largely focused on specific aspects of culture and evaluation. While these efforts contribute to our understanding of cultural competence, a unified and systematic evaluation approach is needed for us as a field to comprehensively assess diverse cultural dimensions at scale. Drawing on measurement theory, we present a principled framework to aggregate multifaceted indicators of cultural capabilities into a unified assessment of cultural intelligence. We start by developing a working definition of culture that includes identifying core domains of culture. We then introduce a broad-purpose, systematic, and extensible framework for assessing cultural intelligence of AI systems. Drawing on theoretical framing from psychometric measurement validity theory, we decouple the background concept (i.e., cultural intelligence) from its operationalization via measurement. We conceptualize cultural intelligence as a suite of core capabilities spanning diverse domains, which we then operationalize through a set of indicators designed for reliable measurement. Finally, we identify the considerations, challenges, and research pathways to meaningfully measure these indicators, specifically focusing on data collection, probing strategies, and evaluation metrics.