We used statistical properties of drainage networks on Mars as a measure of martian landscape morphology and an indicator of landscape evolution processes. We utilize the Mars Orbiter Laser Altimeter (MOLA) data to construct digital elevation maps (DEMs) of several, mostly ancient, martian terrains. Drainage basins and channel networks are computationally extracted from DEMs and their structures are analyzed and compared to drainage networks extracted from terrestrial and lunar DEMs. We show that martian networks are self-affine statistical fractals with planar properties similar to terrestrial networks, but vertical properties similar to lunar networks. The uniformity of martian drainage density is between those for terrestrial and lunar landscapes. Our results are consistent with the roughening of ancient martian terrains by combination of rainfall-fed erosion and impacts, although roughening by other fluvial processes cannot be excluded. The notion of sustained rainfall in recent Mars history is inconsistent with our findings.