We study theoretically light beam propagation in one-dimensional periodic media with intensity-resonant nonlinearity. The phenomenon of discrete modulational instability is investigated in detail as well as the conditions for the existence and stability of fundamental lattice and surface soliton modes. According to the linear stability analysis, only on-site solitons are stable. The mobility of lattice solitons is analyzed by both free energy and mapping concepts. Only broad solitons may freely traverse the lattice. © 2008 The American Physical Society.