We present a semianalytical point-dipole method that uses Ewald lattice summation to find the dispersion relation of guided plasmonic and bianisotropic modes in metasurfaces composed of two-dimensional (2D) periodic lattices of arbitrarily strongly scattering magnetoelectric dipole scatterers. This method takes into account all retarded electrodynamic interactions as well as radiation damping self-consistently. As illustration, we analyze the dispersion of plasmon nanorod lattices, and of 2D split ring resonator lattices. Plasmon nanorod lattices support transverse and longitudinal in-plane electric modes. Scatterers that have an in-plane electric and out-of-plane magnetic polarizability, but without intrinsic magnetoelectric coupling, result in two bands that are mixtures of the bands of electric-only and magnetic-only lattices. Thereby, bianisotropy through mutual coupling, in absence of building-block bianisotropy, is evident. Once strong bianisotropy is included in each building block, the Bloch modes become even more strongly magnetoelectric. Our results are important to understand spatial dispersion and bianisotropy of metasurface and metamaterial designs.