Photoluminescence (PL) properties of type-II InSb/InAs periodic nanostructures containing above-monolayer (ML)-thick InSb insertions, grown by molecular beam epitaxy, are studied by using an FTIR spectrometer in wide temperature range. The samples exhibit bright PL in the 3.5-5.5 mu m range, which is attributed to recombination of holes localized in InSb with electrons accumulated nearby in the InAs matrix. An increase in the InSb nominal thickness from 1 ML to 1.6 ML results in an increase of the PL peak wavelength up to 5.5 mu m (300 K), and significantly improves luminescence intensity at 300 K due to a twice larger energy of hole localization. The InSb/InAs nanostructures also demonstrate an anomalous 'blue' shift of the PL peak energy as the temperature increases in the 12-80 K range, which is attributed to the thermally induced population of localized states in the InSb insertions, emerging due to composition/thickness fluctuations. Sb segregation in the cap InAs barrier smooths the potential inhomogeneities in the insertions, which reduces the broadening parameter.