Requirements are inherently interconnected through various types of dependencies. Identifying these dependencies is essential, as they underpin critical decisions and influence a range of activities throughout software development. However, this task is challenging, particularly in modern software systems, given the high volume of complex, coupled requirements. These challenges are further exacerbated by the ambiguity of Natural Language (NL) requirements and their constant change. Consequently, requirement dependency detection is often overlooked or performed manually. Large Language Models (LLMs) exhibit strong capabilities in NL processing, presenting a promising avenue for requirement-related tasks. While they have shown to enhance various requirements engineering tasks, their effectiveness in identifying requirement dependencies remains unexplored. In this paper, we introduce LEREDD, an LLM-based approach for automated detection of requirement dependencies that leverages Retrieval-Augmented Generation (RAG) and In-Context Learning (ICL). It is designed to identify diverse dependency types directly from NL requirements. We empirically evaluate LEREDD against two state-of-the-art baselines. The results show that LEREDD provides highly accurate classification of dependent and non-dependent requirements, achieving an accuracy of 0.93, and an F1 score of 0.84, with the latter averaging 0.96 for non-dependent cases. LEREDD outperforms zero-shot LLMs and baselines, particularly in detecting fine-grained dependency types, where it yields average relative gains of 94.87% and 105.41% in F1 scores for the Requires dependency over the baselines. We also provide an annotated dataset of requirement dependencies encompassing 813 requirement pairs across three distinct systems to support reproducibility and future research.