We present the first fully relativistic numerical calculations of differentially rotating strange quark stars models for broad ranges of the maximum density and of the degree of differential rotation. Our simulations are performed with the very accurate and stable multi-domain spectral code FlatStar and use the MIT Bag model for describing strange quark matter. Our calculations, based on a thorough exploration of the solution space, show that the maximum mass of strange stars depends on both the degree of differential rotation and a type of solution, similar to neutron stars. The highest increase of the maximum mass (compared to the value for a non-rotating star) is obtained for models with a low degree of differential rotation. This highest mass is over four times larger than that of the equivalent non-rotating configuration. Comparing our results with calculations done for realistic models of neutron stars, we conclude that for small degrees of differential rotation, strange stars can sustain masses much larger than stars made from nuclear matter, which reinforces the hope of demonstrating, or of ruling out, the existence of strange matter through observation of the gravitational waves, gamma-rays, or neutrinos of the massive material object born from the merger of a compact binary system or during some supernova events.