The medium-and high-spin band structure of 103Rh, 104Rh and 105Pd has been studied in order to search for new multiple chiral doublet bands and bands corresponding to the recently predicted transverse wobbling motion. Several new band structures have been identified and their properties compared with constrained covariant density functional theory and particle rotor model calculations. Based on this comparison, three chiral band pairs were identified in 103Rh, of which two belong to the same configuration. Several new positive-and negative-parity bands in 104Rh can form chiral band pairs according to their experimental properties and to the preliminary calculations, thus the existence of multiple chiral doublet bands in this nucleus is possible. Experimental properties of one of the observed negative-parity bands in 105Pd are characteristic of transverse wobbling motion, predicted also by calculations in this nucleus.