Recently we showed that light focusing and transport properties of chains of dielectric spheres with D>>10λ, where D is the diameter of the sphere and λ is the wavelength, are dominated by periodically focused modes (PFMs) which have extremely small propagation losses . In this work we show that along with a special case of PFMs in chains of spheres with index n 3 which propagates in such structures without losses, similar periodic modes exist in a broad range of indices from 1.4 to 2.0. For each n such generalized PFMs have various radial extents in the regions between the neighbouring focused beams. We show that for 10-sphere long chains with 1.68 n 1.80 such modes have total propagation losses smaller than 1 dB. Using numerical ray tracing, we demonstrate that such chains filter radially polarized beams. Using collimated incident beams, a polarization degree in excess of 0.9 is demonstrated for 10-sphere long chains for 1.68 n 1.80 range. These properties make chains of microspheres instrumental for developing novel focusing and polarization components.