SUMMARY We present an operator‐splitting scheme for fluid–structure interaction (FSI) problems in hemodynamics, where the thickness of the structural wall is comparable to the radius of the cylindrical fluid domain. The equations of linear elasticity are used to model the structure, while the Navier–Stokes equations for an incompressible viscous fluid are used to model the fluid. The operator‐splitting scheme, based on the Lie splitting, separates the elastodynamics structure problem from a fluid problem in which structure inertia is included to achieve unconditional stability. We prove energy estimates associated with unconditional stability of this modular scheme for the full nonlinear FSI problem defined on a moving domain, without requiring any sub‐iterations within time steps. Two numerical examples are presented, showing excellent agreement with the results of monolithic schemes. First‐order convergence in time is shown numerically. Modularity, unconditional stability without temporal sub‐iterations, and simple implementation are the features that make this operator‐splitting scheme particularly appealing for multi‐physics problems involving FSI. Copyright © 2013 John Wiley & Sons, Ltd. A novel stable, modular, operator‐splitting scheme is presented for the simulation of fluid‐structure interaction (FSI) problems in which the structure has finite thickness, comparable to the transverse dimension of the fluid domain. The fluid flow is modeled by the Navier‐Stokes equations for an incompressible, viscous fluid, while the structure elastodynamics is governed by the equations of linear elasticity/viscoelasticity. Energy estimates associated with unconditional stability of the scheme are shown for the full, nonlinear FSI problem. Numerical examples show that this modular scheme compares well with monolithic schemes.