The phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices is studied by means of the transfer-matrix method and the real-space renormalization-group transformation. We find that the phase diagram, for the ferromagnetic case, consists of five phases, i.e., the fully disordered paramagnetic phase P, the fully ordered ferromagnetic phase F, and three partially ordered ferromagnetic phases Fs, F s, and Fs s, as well as ten nontrivial fixed points. The correlation length critical exponents and the crossover exponents are also calculated. In addition, we also investigate the variations of the critical exponents with the fractal dimension df, the number of branches m, and the number of bonds per branch b of the generator of the family of diamond-type hierarchical lattices. Finally we give a brief discussion about universality.