SQL query rewriting aims to reformulate a query into a more efficient form while preserving equivalence. Most existing methods rely on predefined rewrite rules. However, such rule-based approaches face fundamental limitations: (1) fixed rule sets generalize poorly to novel query patterns and struggle with complex queries; (2) a wide range of effective rewriting strategies cannot be fully captured by declarative rules. To overcome these issues, we propose using large language models (LLMs) to generate rewrites. LLMs can capture complex strategies, such as evaluation reordering and CTE rewriting. Despite this potential, directly applying LLMs often results in performance regressions or non-equivalent rewrites due to a lack of execution awareness and semantic grounding. To address these challenges, We present E3-Rewrite, an LLM-based SQL rewriting framework that produces executable, equivalent, and efficient queries. It integrates two core components: a context construction module and a reinforcement learning framework. First, the context module leverages execution plans and retrieved demonstrations to build bottleneck-aware prompts that guide inference-time rewriting. Second, we design a reward function targeting executability, equivalence, and efficiency, evaluated via syntax checks, equivalence verification, and cost estimation. Third, to ensure stable multi-objective learning, we adopt a staged curriculum that first emphasizes executability and equivalence, then gradually incorporates efficiency. Across multiple SQL benchmarks, our experiments demonstrate that E3-Rewrite can shorten query execution time by as much as 25.6% relative to leading baselines, while also producing up to 24.4% more rewrites that meet strict equivalence criteria. These gains extend to challenging query patterns that prior approaches could not effectively optimize.