We report on the physical properties of single crystalline EuRhSi3 and polycrystalline EuIrSi3, inferred from magnetization, electrical transport, heat capacity and (151)Eu Mössbauer spectroscopy. These previously known compounds crystallise in the tetragonal BaNiSn3-type structure. The single crystal magnetization in EuRhSi3 has a strongly anisotropic behaviour at 2 K with a spin-flop field of 13 T, and we present a model of these magnetic properties which allows the exchange constants to be determined. In both compounds, specific heat shows the presence of a cascade of two close transitions near 50 K, and the (151)Eu Mössbauer spectra demonstrate that the intermediate phase has an incommensurate amplitude modulated structure. We find anomalously large values, with respect to other members of the series, for the RKKY Néel temperature, for the spin-flop field (13 T), for the spin-wave gap (≃20-25 K) inferred from both resistivity and specific heat data, for the spin-disorder resistivity in EuIrSi3 (≃240 μΩ cm) and for the saturated hyperfine field (52 T). The enhanced values of the quantities that depend on the electronic density of states at the Fermi level, imply that the latter must be strongly enhanced in these two materials. EuIrSi3 exhibits a giant magnetoresistance ratio, with values exceeding 600% at 2 K in a field of 14 T.