We present BYOKG, a universal question-answering (QA) system that can operate on any knowledge graph (KG), requires no human-annotated training data, and can be ready to use within a day -- attributes that are out-of-scope for current KGQA systems. BYOKG draws inspiration from the remarkable ability of humans to comprehend information present in an unseen KG through exploration -- starting at random nodes, inspecting the labels of adjacent nodes and edges, and combining them with their prior world knowledge. In BYOKG, exploration leverages an LLM-backed symbolic agent that generates a diverse set of query-program exemplars, which are then used to ground a retrieval-augmented reasoning procedure to predict programs for arbitrary questions. BYOKG is effective over both small- and large-scale graphs, showing dramatic gains in QA accuracy over a zero-shot baseline of 27.89 and 58.02 F1 on GrailQA and MetaQA, respectively. On GrailQA, we further show that our unsupervised BYOKG outperforms a supervised in-context learning method, demonstrating the effectiveness of exploration. Lastly, we find that performance of BYOKG reliably improves with continued exploration as well as improvements in the base LLM, notably outperforming a state-of-the-art fine-tuned model by 7.08 F1 on a sub-sampled zero-shot split of GrailQA.