Language Models (LMs) are typically tuned with human preferences to produce helpful responses, but the impact of preference tuning on the ability to handle culturally diverse queries remains understudied. In this paper, we systematically analyze how native human cultural preferences can be incorporated into the preference learning process to train more culturally aware LMs. We introduce CARE, a multilingual resource containing 3,490 culturally specific questions and 31.7k responses with human judgments. We demonstrate how a modest amount of high-quality native preferences improves cultural awareness across various LMs, outperforming larger generic preference data. Our analyses reveal that models with stronger initial cultural performance benefit more from alignment, leading to gaps among models developed in different regions with varying access to culturally relevant data. CARE is publicly available at https://github.com/Guochry/CARE.