Coulomb explosion imaging (CEI) with x-ray free electron lasers has recently been shown to be a powerful method for obtaining detailed structural information of gas-phase planar ring molecules [R. Boll et al., X-ray multiphoton-induced Coulomb explosion images complex single molecules, Nat. Phys. 18, 423 (2022).NPAHAX1745-247310.1038/s41567-022-01507-0]. In this Letter, we investigate the potential of CEI driven by a tabletop laser and extend this approach to differentiating three-dimensional structures. We study the static CEI patterns of planar and nonplanar organic molecules that resemble the structures of typical products formed in ring-opening reactions. Our results reveal that each molecule exhibits a well-localized and distinctive pattern in three-dimensional fragment-ion momentum space. We find that these patterns yield direct information about the molecular structures and can be qualitatively reproduced using a classical Coulomb explosion simulation. Our findings suggest that laser-induced CEI can serve as a robust method for differentiating molecular structures of organic ring and chain molecules. As such, it holds great promise as a method for following ultrafast structural changes, e.g., during ring-opening reactions, by tracking the motion of individual atoms in pump-probe experiments.