We study theoretically the linear and nonlinear propagation of light in one-dimensional bi-periodic arrays of fibers, with the propagation constant periodically modulated along the propagation direction. We predict analytically and observe numerically subdiffractive propagation along such fiber arrays, and characterize the light propagation properties. We also predict novel subdiffractive discrete solitons in the presence of Kerr nonlinearity, both of focusing and defocusing cases, which are essentially different from the usual discrete solitons in waveguide arrays.