In this paper, we investigate the spectral properties of condensed matrices in mixed hybrid discretizations of the Neumann boundary value problem for the Poisson equation on distorted meshes. We also consider a new approach to the construction of spectrally equivalent preconditioners to the condensed matrices. The proposed preconditioners are the stiffness matrices of the standard piecewise linear finite element method on triangular/tetrahedral meshes. Generalization to the Dirichlet and mixed boundary conditions as well as to variable coefficients is basically straightforward.