Recent advances in speech deepfake detection (SDD) have significantly improved artifacts-based detection in spoofed speech. However, most models overlook speech naturalness, a crucial cue for distinguishing bona fide speech from spoofed speech. This study proposes naturalness-aware curriculum learning, a novel training framework that leverages speech naturalness to enhance the robustness and generalization of SDD. This approach measures sample difficulty using both ground-truth labels and mean opinion scores, and adjusts the training schedule to progressively introduce more challenging samples. To further improve generalization, a dynamic temperature scaling method based on speech naturalness is incorporated into the training process. A 23% relative reduction in the EER was achieved in the experiments on the ASVspoof 2021 DF dataset, without modifying the model architecture. Ablation studies confirmed the effectiveness of naturalness-aware training strategies for SDD tasks.