We use linear estimators to determine the magnitude and direction of the cosmic radio dipole from the NRAO VLA Sky Survey (NVSS) and the Westerbork Northern Sky Survey (WENSS). We show that special attention has to be given to the issues of bias due to shot noise, incomplete sky coverage and masking of the Milky Way. We compare several different estimators and show that conflicting claims in the literature can be attributed to the use of different estimators. We find that the NVSS and WENSS estimates of the cosmic radio dipole are consistent with each other and with the direction of the cosmic microwave background (CMB) dipole. We find from the NVSS a dipole amplitude of (1.8 ± 0.6) × 10-2 in direction (RA,dec) = (154° ± 19°, −2° ± 19°). This amplitude exceeds the one expected from the CMB by a factor of about 4 and is inconsistent with the assumption of a pure kinetic origin of the radio dipole at 99.6% CL.