We consider the numerical solution of boundary value problems for poroelastic plates. The basic system of equations consists of the biharmonic equation for vertical displacement and nonstationary equation for pressure in the porous medium. The computational algorithm is based on the finite element approximation in longitudinal coordinates and the finite-difference approximation in time. We formulate standard stability conditions for two-level schemes with weights. The computational implementation of such schemes is based on solving a system of coupled equations: fourth-order elliptic equation for displacement and second-order elliptic equation for pressure. We construct unconditionally stable splitting schemes with respect to physical processes, when the transition to a new time level is associated with solving separate problems for the desired displacement and pressure.