While large language models (LLMs) are increasingly used to assist users in various tasks through natural language interactions, these interactions often fall short due to LLMs' limited ability to infer contextual nuances and user intentions, unlike humans. To address this challenge, we draw inspiration from the Gricean Maxims--human communication theory that suggests principles of effective communication--and aim to derive design insights for enhancing human-AI interactions (HAI). Through participatory design workshops with communication experts, designers, and end-users, we identified ways to apply these maxims across the stages of the HAI cycle. Our findings include reinterpreted maxims tailored to human-LLM contexts and nine actionable design considerations categorized by interaction stage. These insights provide a concrete framework for designing more cooperative and user-centered LLM-based systems, bridging theoretical foundations in communication with practical applications in HAI.