Optimum shielding of the radiation from particle accelerators requires knowledge of the attenuation characteristics of the shielding material. The most common material for shielding this radiation is concrete, which can be made using various materials of different densities as aggregates. These different concrete mixes can have very different attenuation characteristics. Information about the attenuation of leakage photons and neutrons in ordinary and heavy concrete is, however, very limited. To increase our knowledge and understanding of the radiation attenuation in concrete of various compositions, we have performed measurements of the transmission of leakage radiation, photons and neutrons, from a Varian Clinac 2100C medical linear accelerator operating at maximum electron energies of 6 and 18 MeV. We have also calculated, using Monte Carlo techniques, the leakage neutron spectra and its transmission through concrete. The results of these measurements and calculations extend the information currently available for designing shielding for medical electron accelerators. Photon transmission characteristics depend more on the manufacturer of the concrete than on the atomic composition. A possible cause for this effect is a non-uniform distribution of the high-density aggregate, typically iron, in the concrete matrix. Errors in estimated transmission of photons can exceed a factor of three, depending on barrier thickness, if attenuation in high-density concrete is simply scaled from that of normal density concrete. We found that neutron transmission through the high-density concretes can be estimated most reasonably and conservatively by using the linear tenth-value layer of normal concrete if specific values of the tenth-value layer of the high-density concrete are not known. The reason for this is that the neutron transmission depends primarily on the hydrogen content of the concrete, which does not significantly depend on concrete density. Errors of factors of two to more than ten, depending on barrier thickness, in the estimated transmission of neutrons through high-density concrete can be made if the attenuation is scaled by density from normal concrete.