A general setting for a new approach to constructing vector splitting schemes in mixed FEM for heat transfer problem is considered. For space approximation Raviart-Thomas finite elements of lowest order are implemented on rectangular (parallelepiped) mesh. The main question discussed is how to implement time discretization so as to obtain efficient numerical algorithms. The key idea of the proposed approach is to use scalar splitting schemes for heat flux divergence. This allows one to carry out accuracy and stability analysis on the basis of the well-known results for underlying scalar splitting schemes. A priori estimates are obtained for vector splitting schemes in two- and three-dimensional cases using the idea of flux decomposition onto divergence-free and potential components at discrete level. It is shown that splitting schemes which have been proposed by other approaches based on Uzawa algorithm's modification and block triangular factorization are included as special cases by an appropriate choice of the underlying splitting schemes for heat flux divergence. Generalization of the discussed approach is presented for second-order hyperbolic problems. Moreover, the general idea of using splitting schemes for the flux divergence to construct splitting schemes for fluxes can be easily extended to first-order hyperbolic systems, but by now several questions still remain unsolved.