Based on asymmetric graphene ellipses, the tunable propagation characteristics of metamaterial absorber (MMA) have been investigated in the THz region. Two distinct absorption peaks of 84% and 90% are observed at 1.06 THz and 1.67 THz, respectively. Besides a high Q factor exceeding 20, the Fano resonance can also be modulated in a wide range (e.g., the frequency modulation depth reaches more than 43.8% if the Fermi energy level changes in the range of 0.2-1.0 eV). Additionally, a bidirectional THz MMA is achieved by replacing the metal substrate with a uniform graphene layer. If the terahertz wave is incident in the forward direction, the proposed graphene double stripe microstructure shows a typical MMA with its absorption reaching 88%. On the other hand, if the terahertz wave is incident in the reverse direction, the graphene double stripe microstructure behaves as a reflective modulator, and its amplitude and frequency MD will reach 60% and 85%. These results contribute to the design of tunable THz devices, such as filters, absorbers, and modulators.