In this paper, we demonstrate a tunable periodic hourglass-shaped graphene arrays absorber in the infrared (IR) and terahertz (THz) frequency bands. The effects of graphene geometric parameters, chemical potentials, periods, and incident angles on the pure absorption characteristics are studied by using the Finite Difference Time Domain (FDTD) method. In addition, this paper also analyzes the pure absorption characteristics of bilayer graphene arrays. The simulation results show that the maximum absorption reaches 38.2% for the monolayer graphene structure. Furthermore, comparing the bilayer graphene structure with the monolayer structure under the same conditions shows that the bilayer structure has a tunable dual-band selective absorption effect and has a higher maximum absorption of 41.7%. Moreover, it was found that there are dual-band tunable absorption peaks at 26 μm and 36.3 μm with the maximum absorption of 41.7% and 11%. The proposed structure is a convenient method which could be used in the design of graphene-based optoelectronic devices, biosensors, and environmental monitors.