We have studied the single-particle properties and the equation of state (EOS) of symmetric nuclear and neutron matter within the framework of the Brueckner-Hartree-Fock (BHF) approach extended by including a phenomenological three-body force (3BF). Adding the 3BF to the initial two-body force (2BF) and applying a partial-wave expansion, G-matrix calculations are performed in pure neutron matter as well as in symmetric nuclear matter. The 3BF is shown to affect significantly the nuclear matter EOS at high densities above the normal nuclear matter density and it is necessary for reproducing the empirical saturation property of symmetric nuclear matter in a non-relativistic microscopic framework. The calculations have been done using the charge-dependent chiral nucleon-nucleon interaction at order four N3LO potential and compared with both the CD-Bonn and Argonne V18 potentials plus the three-nucleon Urbana interaction. It is found that the calculations with the N3LO potential alone for symmetric nuclear matter at high density is strongly over bound in the BHF approximation.