This paper extends the unfitted Nitsche-XFEM method of [Comput. Methods Appl. Mech. Engrg., 301, 300-335, 2016] to three-dimensional fluid-structure interaction problems with immersed thin-walled elastic solids. The fluid and solid domains are discretized with unfitted unstructured meshes. Discrete weak and strong discontinuities are allowed in the fluid and the coupling is enforced consistently via a fluid-sided Nitsche's type mortaring with suitable stabilization for robustness. Integration over cut-elements is handled via an efficient and robust intersection and subtesselation algorithm. The method includes a new approach for the treatment of partially intersected fluid domains. Several numerical examples are presented and discussed, which illustrate the capabilities of the proposed method.