Being motivated by the recent progress in attosecond laser technology, we theoretically explore the strategy of inducing ultrafast electron dynamics inherent to aromatic molecules, i.e., ring currents by means of polarized laser pulses. The main topic of discussion is how to control the direction of ring currents in an aromatic molecule of low symmetry, for which the design of an efficient control pulse cannot be achieved intuitively. We first consider a system with a single aromatic ring and show that coherent π-electron angular momentum, which oscillates with time, can be produced and controlled by a polarized laser pulse with its ellipticity and orientation properly chosen. Nonadiabatic couplings with molecular vibration gradually weaken the angular momentum, while the vibrational amplitude strongly depends on the polarization of incident light. This suggests the conversion of the polarization dependence of ring current into that of subsequent vibration, which may open a way to detect laser-driven ultrafast electron dynamics by vibrational spectroscopy. The laser-control scheme for the ring current is then extended to a molecule with two aromatic rings, which exhibits characteristic phenomena absent in that with a single ring. We demonstrate that two-dimensional switching of the direction of angular momentum is possible in such molecules. In addition, ring current can be localized at a specific ring by tailored lasers. The application of the present control method to polycyclic aromatic hydrocarbons will lead to the development of next-generation organic optical switching devices.