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Former studies have found that water droplets on the rice leaves exhibit anisotropic sliding behavior, which is mainly caused by the anisotropic groove-like surface microstructures. Similar groove-like surface microstructures and anisotropic sliding behavior of water droplets can also be found on some other plants, such as the reed leaves. In this paper, biomimetic groove-like surface microstructures are fabricated on copper surfaces by femtosecond laser micromachining. Thanks to the flexibility of this method, the period and height of the periodic micro-grooves can be precisely controlled, and the nanoscale surface roughness can also be regulated. The influences of these surface morphologies on the wetting state and anisotropic sliding behavior of water droplets are studied systematically, and theoretic analyses are also conducted. Our results prove that the anisotropic sliding behavior is mainly decided by the wetting state of external water droplets. Besides, plenty of surface nanostructures have no benefits to the anisotropic sliding behavior of surface water droplets.