Computational efficiency remains a significant challenge in the development of effective strate- gies for simulating complex, industrially relevant problems involving fluid-structure interaction. Despite recent advancements, there is still a significant need for methods that effectively bal- ance accuracy, stability, and computational cost. This work introduces new staggered schemes designed with optimal predictors based on Dirichlet-Neumann coupling. These schemes are second-order accurate and maintain unconditional stability up to a critical amount of added mass. To further improve the accuracy of these schemes, a non-iterative strategy is introduced, in which a predefined number of computational cycles is executed per time step. This approach simplifies implementation and significantly reduces computational costs compared to classical iterative methods, which often require a large number of iterations per time step to converge. It exploits and further develops the ideas presented in [1-3] . The properties and performance of the proposed schemes are thoroughly evaluated through a linear model problem derived from a simplified fluid-conveying elastic tube. The strategies are evaluated through several benchmark problems in both two-dimensional and three-dimensional fluid-structure interaction problems, which are designed to push the schemes to their stability limits in order to assess their performance under strongly coupled interactions.