We study when large language models (LLMs) can serve as effective black-box policy optimizers for reinforcement learning (RL) tasks, i.e., when can we replace classical RL algorithms with an LLM? We explore this question by introducing Prompted Policy Optimization (PromptPO), an iterative method that prompts an LLM with Python descriptions of the state space, action space, and reward function, then has it generate and refine executable policies based on rollout feedback. Across hard exploration environments, Meta-World robotics tasks, and several real-world control problems, PromptPO often matches or exceeds the performance of standard RL baselines while using substantially fewer environment interactions. To maximize expected return, and without further explicit prompting, the policies PromptPO outputs range from tuned proportional controllers or rule-based plans to policies that run planning algorithms like value iteration. Our results demonstrate that LLM-based policy optimization is sufficient when the LLM can leverage prior knowledge about the environment or optimization strategy. PromptPO underperforms standard RL baselines in MuJoCo domains. This demonstrates possible limitations of LLM-based policy optimization to settings that requiring fine-grained continuous control.