Clinical decision-making in radiology increasingly benefits from artificial intelligence (AI), particularly through large language models (LLMs). However, traditional retrieval-augmented generation (RAG) systems for radiology question answering (QA) typically rely on single-step retrieval, limiting their ability to handle complex clinical reasoning tasks. Here we propose radiology Retrieval and Reasoning (RaR), a multi-step retrieval and reasoning framework designed to improve diagnostic accuracy, factual consistency, and clinical reliability of LLMs in radiology question answering. We evaluated 25 LLMs spanning diverse architectures, parameter scales (0.5B to >670B), and training paradigms (general-purpose, reasoning-optimized, clinically fine-tuned), using 104 expert-curated radiology questions from previously established RSNA-RadioQA and ExtendedQA datasets. To assess generalizability, we additionally tested on an unseen internal dataset of 65 real-world radiology board examination questions. RaR significantly improved mean diagnostic accuracy over zero-shot prompting and conventional online RAG. The greatest gains occurred in small-scale models, while very large models (>200B parameters) demonstrated minimal changes (<2% improvement). Additionally, RaR retrieval reduced hallucinations (mean 9.4%) and retrieved clinically relevant context in 46% of cases, substantially aiding factual grounding. Even clinically fine-tuned models showed gains from RaR (e.g., MedGemma-27B), indicating that retrieval remains beneficial despite embedded domain knowledge. These results highlight the potential of RaR to enhance factuality and diagnostic accuracy in radiology QA, warranting future studies to validate their clinical utility. All datasets, code, and the full RaR framework are publicly available to support open research and clinical translation.