This work proposes a unique configuration of a two-dimensional metamaterial lattice grid comprising curved and tapered beams. The propagation of elastic waves in the structure is analyzed using the dynamic stiffness matrix (DSM) approach and the Floquet–Bloch theorem. The DSM for the unit cell is formulated under the extensional theory of curved beam, considering the effects of shear and rotary inertia. The study considers two types of variable rectangular cross sections, viz. single taper and double taper along the length of the beam. Further, the effect of curvature and taper on the wave propagation is analyzed through the band diagram along the irreducible Brillouin zone. It is shown that a complete band gap, i.e., attenuation band in all the directions of wave propagation, in a homogeneous structure can be tailored with a suitable combination of curvature and taper. Generation of the complete bandgap is hinged upon the coupling of the axial and transverse components of the lattice grid. This coupling emerges due to the presence of the curvature and is further enhanced due to tapering. The double taper cross section is shown to have wider attenuation characteristics than single taper cross sections. Specifically, 83.36% and 63% normalized complete bandwidth is achieved for the double and single taper cross section for a homogeneous metamaterial, respectively. Additional characteristics of the proposed metamaterial in the time and frequency domain of the finite structure, vibration attenuation, wave localization in the equivalent finite structure are also studied.