The growing need of fine-tuning large language models (LLMs) can lead to significant energy consumption and environmental impact. To address this issue, we introduce GreenTrainer, a novel LLM fine-tuning technique. GreenTrainer assesses the backpropagation costs and contributions of different tensors to model accuracy, allowing for the selection of the most efficient set of tensors. This selection is guided by a user-defined objective, which can adapt to energy supply considerations and Green AI goals. Experimental results demonstrate that GreenTrainer can reduce FLOPs by up to 64% without compromising model accuracy, and outperforms existing techniques like LoRA while maintaining comparable FLOPs reduction.