In this work, we propose and analyse a new scheme to discretize the linearized version of a rather general poromechanics model adapted to biological tissues perfusion. This model, which is related to – albeit different from – Biot equations, involves unsteady solid and fluid momentum balance equations that are further coupled through an incompressibility constraint, a pore pressure and permeability terms. The key feature of the scheme is to decouple the solid, fluid and pressure unknowns at each time step by means of a projection method, composed of a prediction and a correction step. We perform a complete stability analysis of the scheme depending on the implicit or explicit treatment of friction and pressure in the prediction step. Several boundary conditions are considered, including conditions coupling the solid and fluid phases on the boundary that are imposed at the discrete level using a Robin-Robin method. In the case of Dirichlet boundary conditions, we also provide a fully discrete error estimate as long as a discrete inf-sup condition is satisfied. The scheme properties and robustness with respect to physical parameters are illustrated by numerical experiments. Finally, its computational performance is compared with that of a monolithic approach.