Long, narrow grooves found on the slopes of martian sand dunes have been cited as evidence of liquid water via the hypothesis that melt-water initiated debris flows eroded channels and deposited lateral levées. However, this theory has several short-comings for explaining the observed morphology and activity of these linear gullies. We present an alternative hypothesis that is consistent with the observed morphology, location, and current activity: that blocks of CO₂ ice break from over-steepened cornices as sublimation processes destabilize the surface in the spring, and these blocks move downslope, carving out levéed grooves of relatively uniform width and forming terminal pits. To test this hypothesis, we describe experiments involving water and CO₂ blocks on terrestrial dunes and then compare results with the martian features. Furthermore, we present a theoretical model of the initiation of block motion due to sublimation and use this to quantitatively compare the expected behavior of blocks on the Earth and Mars. The model demonstrates that CO₂ blocks can be expected to move via our proposed mechanism on the Earth and Mars, and the experiments show that the motion of these blocks will naturally create the main morphological features of linear gullies seen on Mars.