Anisotropic materials provide a new platform for building diverse polarization-dependent optical devices. Two-dimensional α-phase molybdenum trioxides (α-MoO3), as newly emerging natural van der Waals materials, have attracted significant attention due to their unique anisotropy. In this work, we theoretically propose an anisotropic perfect metamaterial absorber in visible frequencies, the unit cell of which consists of a multi-layered α-MoO3 nanoribbon/dielectric structure stacked on a silver substrate. Additionally, the number of perfect absorption bands is closely related to the α-MoO3 nanoribbon/dielectric layers. When the proposed absorber is composed of three α-MoO3 nanoribbon/dielectric layers, electromagnetic simulations show that triple-band perfect absorption can be achieved for polarization along [100], and [001] in the direction of, α-MoO3, respectively. Moreover, the calculation results obtained by the finite-difference time-domain (FDTD) method are consistent with the effective impedance of the designed absorber. The physical mechanism of multi-band perfect absorption can be attributed to resonant grating modes and the interference effect of Fabry–Pérot cavity modes. In addition, the absorption spectra of the proposed structure, as a function of wavelength and the related geometrical parameters, have been calculated and analyzed in detail. Our proposed absorber may have potential applications in spectral imaging, photo-detectors, sensors, etc.