We investigate theoretically the band structure of flexural waves propagating in an elastic metamaterial thin plate. Kirchhoff-Love thin plate theory is considered. We study the influence of periodic arrays of multiple degrees of freedom local resonators in square and triangular lattices. Plane wave expansion and extended plane wave expansion methods, also known as ω(k) and k(ω), respectively, are used to solve the governing equation of motion for a thin plate. The locally resonant band gaps for square and triangular lattices present almost the same attenuation for all examples analysed. However, square lattice presents broader Bragg-type band gaps with higher attenuation than triangular lattice. An experimental analysis is conducted with a real elastic metamaterial thin plate with resonators in a square lattice. Modal analysis and forced response are computed by finite element method. Plane wave expansion, finite element and experimental results present good agreement.