We demonstrate that a broadband terahertz absorber with near-unity absorption can be realized using a net-shaped periodically sinusoidally-patterned graphene sheet, placed on a dielectric spacer supported on a metallic reflecting plate. Because of the gradient width modulation of the unit graphene sheet, continuous plasmon resonances can be excited, and therefore broadband terahertz absorption can be achieved. The results show that the absorber's normalized bandwidth of 90% terahertz absorbance is over 65% under normal incidence for both TE and TM polarizations when the graphene chemical potential is set as 0.7 eV. And the broadband absorption is insensitive to the incident angles and the polarizations. The peak absorbance remains more than 70% over a wide range of the incident angles up to 60° for both polarizations. Furthermore, this absorber also has the advantage of flexible tunability via electrostatic doping of graphene sheet, which peak absorbance can be continuously tuned from 14% to 100% by controlling the chemical potential from 0 eV to 0.8 eV. The design scheme is scalable to develop various graphene-based tunable broadband absorbers at other terahertz, infrared, and visible frequencies, which may have promising applications in sensing, detecting, and optoelectronic devices.