The electrical properties of organic field-effect transistors (OFETs) are usually characterized by applying models initially developed for inorganic-based devices, which often implies the use of approximations that might be inappropriate for organic semiconductors. These approximations have brought limitations to the understanding of the device physics associated with organic materials. A strategy to overcome this issue is to establish straightforward connections between the macroscopic current characteristics and microscopic charge transport in OFETs. Here, a 3D kinetic Monte Carlo model is developed that goes beyond both the conventional assumption of zero channel thickness and the gradual channel approximation to simulate carrier transport and current. Using parallel computing and a new algorithm that significantly improves the evaluation of electric potential within the device, this methodology allows the simulation of micrometer-sized OFETs. The current characteristics of representative OFET devices are well reproduced, which provides insight into the validity of the gradual channel approximation in the case of OFETs, the impact of the channel thickness, and the nature of microscopic charge transport.