The short-range organization around Ni atoms in orthorhombic RNiO3 (R=Pr, Nd, Eu) perovskites has been studied over a wide temperature range by Ni K-edge x-ray-absorption spectroscopy. Our results demonstrate that two different Ni sites, with different average Ni-O bond lengths, coexist in those orthorhombic compounds and that important modifications in the Ni nearest-neighbors environment take place across the metal-insulator transition. We report evidences for the existence of short-range charge order in the insulating state, as found in the monoclinic compounds. Moreover, our results suggest that the two different Ni sites coexist even in the metallic state. The coexistence of two different Ni sites, independently on the R ion, provides a common ground to describe these compounds and sheds interesting light in the understanding of the phonon-assisted conduction mechanism and unusual antiferromagnetism present in all RNiO3 compounds.