There is a recurring question in solar physics about whether or not electric currents are neutralized in active regions (ARs). This question was recently revisited using three-dimensional (3D) magnetohydrodynamic (MHD) numerical simulations of magnetic flux emergence into the solar atmosphere. Such simulations showed that flux emergence can generate a substantial net current in ARs. Another source of AR currents are photospheric horizontal flows. Our aim is to determine the conditions for the occurrence of net vs. neutralized currents with this second mechanism. Using 3D MHD simulations, we systematically impose line-tied, quasi-static, photospheric twisting and shearing motions to a bipolar potential magnetic field. We find that such flows: (1) produce both {\it direct} and {\it return} currents, (2) induce very weak compression currents - not observed in 2.5D - in the ambient field present in the close vicinity of the current-carrying field, and (3) can generate force-free magnetic fields with a net current. We demonstrate that neutralized currents are in general produced only in the absence of magnetic shear at the photospheric polarity inversion line - a special condition rarely observed. We conclude that, as magnetic flux emergence, photospheric flows can build up net currents in the solar atmosphere, in agreement with recent observations. These results thus provide support for eruption models based on pre-eruption magnetic fields possessing a net coronal current. ; Comment: 14 pages and 11 figures (Accepted in The Astrophysical Journal)