In this paper we propose a simple technique for tetrahedral mesh improvement without inserting Steiner vertices, concentrating mainly on boundary conforming meshes. The algorithm makes local changes to the mesh to remove tetrahedra which are poor according to some quality criterion. While the algorithm is completely general with regard to quality criterion, we target improvement of the dihedral angle. The central idea in our algorithm is the introduction of a new local operation called multi-face retriangulation (MFRT) which supplements other known local operations. Like in many previous papers on tetrahedral mesh improvement, our algorithm makes local changes to the mesh to reduce an energy measure which reflects the quality criterion. The addition of our new local operation allows us to advance the mesh to a lower energy state in cases where no other local change would lead to a reduction. We also make use of the edge collapse operation in order to reduce the size of the mesh while improving its quality. With these operations, we demonstrate that it is possible to obtain a significantly greater improvement to the worst dihedral angles than using the operations from the previous works, while keeping the mesh complexity as low as possible.