In recent decades, origami has transitioned from a traditional art form into a systematic field of scientific inquiry, characterized by attributes such as high foldability, lightweight frameworks, diverse deformation modes, and limited degrees-of-freedom. Despite the abundant literature on smart materials, actuation methods, design principles, and manufacturing techniques, comprehensive reviews focusing on the mechanical properties of origami-inspired structures are relatively rare and unsystematic. This review aims to fill this void by analyzing and summarizing the significant studies conducted on the mechanical properties of origami-inspired structures from 2013 to 2023. We begin with an overview that includes essential definitions of origami, classical origami patterns, and their associated tessellated or stacked structures. Following this, we delve into the principal dynamic modeling method for origami and conduct an in-depth analysis of the key mechanical properties of origami-inspired structures. These properties include tunable stiffness, bistability and multistability, metamechanical properties demonstrated by origami-based metamaterials, and bio-inspired mechanical properties. Finally, we conclude with a comprehensive summary that discusses the current challenges and future directions in the field of origami-inspired structures. Our review provides a thorough synthesis of both the mechanical properties and practical applications of origami-inspired structures, aiming to serve as a reference and stimulate further research.