Gamma rays from the γ-ray burst (GRB) 021206 have been reported to be strongly linearly polarized1, with the estimated degree of polarization (80 ± 20%) being close to the absolute maximum of 100% — affording us the opportunity to constrain models of quantum gravity, which has had 1010 years to act on the photons as they travelled towards us. Here I show that if the effects of quantum gravity are linearly proportional to the ratio of the photon energy to the characteristic scale energy of quantum gravity, then the polarization of photons with energies of about 0.1 MeV should be completely random, contrary to what is observed. I conclude that, should the polarization measurement be confirmed, quantum gravity effects act with a power that is greater than linearity, or that loop quantum gravity is not viable. Compared with previous methods and results (see ref. 2, for example), testing of the linear polarization of cosmic γ-ray bursts may substantially extend the observational window on the theory of quantum gravity.