Claim verification plays a crucial role in combating misinformation. While existing works on claim verification have shown promising results, a crucial piece of the puzzle that remains unsolved is to understand how to verify claims without relying on human-annotated data, which is expensive to create at a large scale. Additionally, it is important for models to provide comprehensive explanations that can justify their decisions and assist human fact-checkers. This paper presents First-Order-Logic-Guided Knowledge-Grounded (FOLK) Reasoning that can verify complex claims and generate explanations without the need for annotated evidence using Large Language Models (LLMs). FOLK leverages the in-context learning ability of LLMs to translate the claim into a First-Order-Logic (FOL) clause consisting of predicates, each corresponding to a sub-claim that needs to be verified. Then, FOLK performs FOL-Guided reasoning over a set of knowledge-grounded question-and-answer pairs to make veracity predictions and generate explanations to justify its decision-making process. This process makes our model highly explanatory, providing clear explanations of its reasoning process in human-readable form. Our experiment results indicate that FOLK outperforms strong baselines on three datasets encompassing various claim verification challenges. Our code and data are available.