In this paper we introduce a class of fully decoupled time-marching schemes (velocity-pressure-displacement splitting) for the coupling of an incompressible fluid with a thin-walled viscoelastic structure. The time splitting combines an overall fractional-step time-marching of the system with a specific Robin-Neumann treatment of the interface coupling. The resulting solution procedures are genuinely partitioned. A priori energy estimates guaranteeing unconditional stability are established for the variants without extrapolation and with first-order extrapolation. The accuracy and performance of the methods proposed are discussed in several numerical examples.