We present a model for determining the threshold voltage of field-effect transistors with nanocrystalline active channel layers. In this type of device, the multiple boundaries between neighboring crystalline grains limit the charge-carrier transport. Electrons in the channel may either populate the conduction band within a grain or be trapped at an interface between neighboring grains. The relative distribution of the electrons over these states determines the conductances of the grains and of the boundaries between them. We employ simple carrier statistics to calculate the macroscopic densities of free and trapped carriers, and these densities are then used to define site and bond occupation probabilities for a two-dimensional site-bond percolation problem. The dependence of the threshold voltage on the primary model parameters: the energy of the trap states, the total density of traps, and the temperature, is explored.