Large language models (LLMs) augmented with knowledge graphs (KGs) offer a promising approach for knowledge-intensive reasoning. Central to this approach is the selection of appropriate reasoning paths in the KG. Yet, existing methods face a common limitation: reasoning path selection is often performed by separate modules using criteria that are only weakly connected to the reasoning requirements. This often results in selecting incorrect relations or premature pruning of relevant paths. We propose Search-on-Graph (SoG), a method that strengthens the connection between path selection and reasoning by having the LLM itself select which relations to follow, informed by both the available KG structure and the complete reasoning history. SoG follows an \textit{observe-think-navigate} paradigm: at each step, the LLM observes the relational connections available at the current entity, reasons about which path best advances toward answering the question, and navigates accordingly. This context-aware navigation fully exploits the LLM's reasoning capabilities rather than relying on independent selection modules with surrogate criteria. Experiments on six knowledge graph question answering (KGQA) benchmarks demonstrate that SoG outperforms state-of-the-art methods while requiring no task-specific fine-tuning and generalizing across different KG schemas.