Large language models (LLMs) have transformed machine translation, yet mistranslations, hallucinations, and unnatural phrasing still limit their effectiveness, particularly for low-resource languages. We propose Translation-CoT, a chain-of-thought prompting strategy that breaks translation into structured stages (lexical retrieval, grammatical analysis, and topic identification), followed by a refinement step to improve fluency, tone, and idiomatic expression. We evaluate Translation-CoT across 14 languages from 14 language families and multiple LLMs (GPT-4o, GPT-4o-mini, LLaMA 3.1, and Gemma 2), with GPT-4o performing best overall, in both English ↔ non-English (X) translation settings. Compared with zero-shot prompting, in-context learning, and existing chain-of-thought prompting methods (Tree-of-Thought (ToT) and Learning-Oriented Prompting (LOT)), Translation-CoT outperforms these prompting strategies on multilingual machine translation across BLEU, ChrF, and METEOR, with especially strong gains in the more difficult English→non-English (X) setting and in low-resource languages. Human evaluation further shows higher preference scores and lower MQM penalty scores, indicating fewer mistranslations, omissions, awkward phrasing, and hallucinations with Translation-CoT. Overall, our results show that structured, task-aware prompting is an effective approach for improving multilingual translation quality and robustness in LLMs.