In this paper, we study triharmonic CMC Lorentz hypersurfaces \(M_{1}^{4}\) in a Lorentz space form \(N_{1}^{5}(c)\) . Generally speaking, the shape operator A on a Lorentz hypersurface \(M_{1}^{4}\) may not be diagonalizable making the investigation on the geometric structures of Lorentz hypersurfaces more complicated. After carefully analyzing the structure equations, we will show that \(\textrm{Tr} A^2\) must be constant. Utilizing this, we are able to prove that any triharmonic CMC Lorentz hypersurface with the shape operator A taking the canonical form (I), (II) or (III) in anti-de Sitter space \({\mathbb {H}}_{1}^{5}\) or Minkowski space \({\mathbb {E}}_1^5\) is minimal; any triharmonic CMC Lorentz hypersurface with the shape operator A taking the canonical form \((\textrm{IV})\) and \(\textrm{Tr} A^2\ge 0 \) in anti-de Sitter space \({\mathbb {H}}_{1}^{5}\) or Minkowski space \({\mathbb {E}}_1^5\) is minimal.
