This work concerns the linearization of a three-field discretization of generalized Biot's equations describing coupled fluid flow and mechanical deformation in unsaturated porous media. The model of interest employs the effective stress based on the so-called equivalent pore pressure and can be interpreted as linear mechanics nonlinearly coupled with Richards' equation. As linearization, we apply simultaneously the L-scheme and the Fixed Stress Splitting scheme, which have been established and analyzed for Richards' equation and the linear Biot's equations, respectively. Numerical results demonstrate robustness and mesh independent convergence rates, whereas the popular, locally convergent Newton's method does not display robust convergence for the numerical examples we present.