Large Language Models (LLMs) have demonstrated remarkable capabilities across various tasks, yet they often exhibit a specific cultural bias, neglecting the values and linguistic diversity of low-resource regions. This cultural bias not only undermines universal equality but also risks reinforcing stereotypes and perpetuating discrimination. To address this, we propose CulFiT, a novel culturally-aware training paradigm that leverages multilingual data and fine-grained reward modeling to enhance cultural sensitivity and inclusivity. Our approach synthesizes diverse cultural-related questions, constructs critique data in multiple culturally relevant languages, and employs fine-grained rewards to decompose cultural texts into verifiable knowledge units for interpretable evaluation. We also introduce GlobalOpinionQA, a multilingual open-ended question-answering dataset designed to evaluate culturally-aware responses in a global context. Extensive experiments on three existing benchmarks and our GlobalOpinionQA demonstrate that CulFiT achieves state-of-the-art open-source model performance in cultural alignment and general reasoning.