The main topic of the paper is to present a way of constructing the second order finite-volume approximations on nonuniform grids to solve 3D mixed boundary value problems for diffusion equation with piecewise constant coefficients. For obtaining the difference equations, a linear combination of the balance relations for the normal flow densities over two boxes is approximated. A set of 19- and 27-point schemes is described and investigated. Representation of the entries of the local balance matrix and assembling of the global balance matrix are given. Monotonicity conditions are obtained in the form of inequalities for meshsteps. The numerical solution error is estimated in the uniform and weighted Eucledian norms. The theoretical approach is confirmed by the results of computational experiments.