The Skyrme-type energy density functionals (EDF) are very widely used for building mean field models of nuclear matter and finite nuclei. Their parameters are usually directly determined by fitting nuclear observables. In this work we attempt to obtain these parameters by relating them to Brueckner-Hartree-Fock (BHF) results obtained in infinite nuclear matter with realistic two-body and three-body forces. These ab initio constraints are supplemented by additional constraints on binding energies of selected nuclei. In this way we can build new Skyrme-type EDF which have a microscopic, ab initio foundation. These EDF can describe properties of finite nuclei (ground states and giant resonances) at a quantitative level comparable with that of phenomenological Skyrme forces.