3D visual grounding is a critical skill for household robots, enabling them to navigate, manipulate objects, and answer questions based on their environment. While existing approaches often rely on extensive labeled data or exhibit limitations in handling complex language queries, we propose LLM-Grounder, a novel zero-shot, open-vocabulary, Large Language Model (LLM)-based 3D visual grounding pipeline. LLM-Grounder utilizes an LLM to decompose complex natural language queries into semantic constituents and employs a visual grounding tool, such as OpenScene or LERF, to identify objects in a 3D scene. The LLM then evaluates the spatial and commonsense relations among the proposed objects to make a final grounding decision. Our method does not require any labeled training data and can generalize to novel 3D scenes and arbitrary text queries. We evaluate LLM-Grounder on the ScanRefer benchmark and demonstrate state-of-the-art zero-shot grounding accuracy. Our findings indicate that LLMs significantly improve the grounding capability, especially for complex language queries, making LLM-Grounder an effective approach for 3D vision-language tasks in robotics. Videos and interactive demos can be found on the project website.