We present expressions for the Green's tensors in terms of lattice sums for general two–dimensional arrays in both elastostatics and elastodynamics. We represent the lattice sums in rapidly convergent forms by Fourier transform methods. We compare the static lattice sums with recently published highly accurate values for square arrays. We also establish Rayleigh identities for elastostatic and elastodynamic problems, generalizing previous work in electrostatics and electromagnetism. The Rayleigh identities may be used in studies of phononic band gaps.