Poroelasticity has a wide range of applications in biology, filtration, tissue engineering and soil sciences. It represents a model for problems where an elastic porous solid is saturated by a viscous fluid. The poroelasticity equations were derived by Biot in 1941, studying the consolidation of soils. In this talk we will present an efficient solution method for the poroelasticity system. A staggered grid discretization is adopted from incompressible flow and modified for the poroelasticity system. Standard finite elements (or finite differences) do not lead to stable solutions without additional stabilization. The staggered grid discretization leads to a natural stable discretization. In contrast to our previous work [1,2], where we provided multigrid solvers for the whole system of poroelasticity equations, here we first split the system into scalar equations. This splitting can be interpreted as a segregated solution approach, similarly to pressure-correction methods in incompressible fluid flow simulation. In fact, we just reverse loops as compared to our previous work: The distributive smoothing method from [1] now acts as the "outer loop" in the solution process. The method can also be seen as a form of preconditioning of the original poroelasticity system. Next to a right- we also present the left-preconditioned system and corresponding results. In the segregated framework we need to solve for scalar equations only, thus enabling the solution of three-dimensional problems on relatively fine grids. Highly efficient multigrid schemes are used to solve the resulting scalar equations. Next to numerical experiments and analysis results we will present some theoretical convergence results for the approach adopted.