Despite their remarkable natural language understanding capabilities, Large Language Models (LLMs) have been underutilized for retrieval tasks. We present Search-R3, a novel framework that addresses this limitation by adapting LLMs to generate search embeddings as a direct output of their reasoning process. Our approach exploits LLMs' chain-of-thought capabilities, allowing them to produce more effective embeddings by reasoning step-by-step through complex semantic analyses. We implement this through three complementary mechanisms. (1) a supervised learning stage enables the model's ability to produce quality embeddings, (2) a reinforcement learning (RL) methodology that optimizes embedding generation alongside reasoning, and (3) a specialized RL environment that efficiently handles evolving embedding representations without requiring complete corpus re-encoding at each training iteration. Our extensive evaluations on diverse benchmarks demonstrate that Search-R3 significantly outperforms prior methods by unifying the reasoning and embedding generation processes. This integrated post-training approach represents a substantial advancement in handling complex knowledge-intensive tasks that require both sophisticated reasoning and effective information retrieval. Project page: https://github.com/ytgui/Search-R3