Locomotion plays a crucial role in shaping the user experience within virtual reality environments. In particular, hands-free locomotion offers a valuable alternative by supporting accessibility and freeing users from reliance on handheld controllers. To this end, traditional speech-based methods often depend on rigid command sets, limiting the naturalness and flexibility of interaction. In this study, we propose a novel locomotion technique powered by large language models (LLMs), which allows users to navigate virtual environments using natural language with contextual awareness. We evaluate three locomotion methods: controller-based teleportation, voice-based steering, and our language model-driven approach. Our evaluation combines eye-tracking data analysis, including exploratory explainable machine learning analysis with SHAP, and standardized questionnaires (SUS, IPQ, CSQ-VR, NASA-TLX) to examine user experience through both objective gaze-based measures and subjective self-reports of usability, presence, cybersickness, and cognitive load. Our findings show no statistically significant differences in usability, presence, or cybersickness between LLM-driven locomotion and established methods such as teleportation, suggesting its potential as a viable, natural language-based, hands-free alternative. In addition, eye-tracking analysis revealed patterns suggesting tendency toward increased user attention and engagement in the LLM-driven condition. Complementary to these findings, exploratory SHAP analysis revealed that fixation, saccade, and pupil-related features vary across techniques, indicating distinct patterns of visual attention and cognitive processing. Overall, we state that our method can facilitate hands-free locomotion in virtual spaces, especially in supporting accessibility.