I consider models in which non-standard SUSY breaking terms, including Dirac gaugino masses, arise from F-term breaking mediated by operators with a cubic mass suppression. In these models, the supersoft properties found in the case of D-term breaking are absent in general, but can be obtained as a special case that is a fixed point of the renormalization group equations. The mu term is replaced by three distinct SUSY-breaking parameters, decoupling the Higgs scalar potential from the Higgsino masses. The complete set of non-standard SUSY-breaking terms with minimal flavor violation are induced in the supersymmetric Standard Model Lagrangian. I will comment briefly on the possible LHC implications.