The efficient solution of fluid-structure interaction (FSI) problems poses significant challenges due to the strong coupling between two heterogeneous physics, especially in regimes where fluid and solid densities are comparable, such as hemodynamics. The associated added-mass effect can severely deteriorate the stability of loosely coupled (LC) partitioned schemes (Causin et al., 2005) -- where only one fluid and one structure problem are solved per time step -- otherwise attractive for their modularity and computational efficiency over strongly coupled (SC) or monolithic approaches, especially in large-scale applications. In this work, we build on the interpretation of the standard Dirichlet-Neumann (DN) algorithm as a Richardson method with a block Gauss-Seidel preconditioner and acceleration parameter alpha = 1, to develop a new coupling strategy that yields a stable scheme under high added-mass conditions. Specifically, we consider the SC method associated with an arbitrary value of alpha (DN-alpha algorithm), which introduces correction terms that improve the convergence of the standard DN method. We analytically prove that the DN-alpha scheme applied to the linear problem proposed in Causin et al. (2005) is convergent for a specific range of alpha without the need for relaxation. Secondly, starting from the DN-alpha scheme, we propose a new LC scheme (DN-alpha-LC), performing a single preconditioned Richardson iteration per time step. We prove that the proposed LC scheme is stable under a constraint on the time step and the Richardson parameter alpha, depending on the fluid-to-structure density ratio. Numerical experiments in realistic settings confirm the theoretical results and show that the DN-alpha-LC solution converges to the SC DN-alpha scheme as the time step is refined. Such results confirm the effectiveness and applicability of the proposed schemes in large-scale cardiovascular simulations. These results highlight the potential of the scheme as a prototype for developing robust preconditioning strategies with respect to physical parameters such as the density ratio.