A comparison is made between two-dimensional elastic discrete lattices and a corresponding Cosserat continuum. Firstly, it is demonstrated how the equations of motion of the Cosserat model can be retrieved from those of a lattice model. For this purpose, two lattice geometries are elaborated: a 7-cell hexagonal lattice and a 9-cell square lattice. For both lattices, the individual cells have two translational spring interactions and a rotational spring interaction with their neighbouring cells. Secondly, the dispersion relations for the lattice models and the Cosserat continuum model are examined in order to determine up to which wavelength of the deformation field the Cosserat model accurately represents the underlying discrete micro-structure. The effect of the lattice anisotropy and inhomogeneity on this accuracy is also discussed.