The current-voltage characteristics of layered organic crystals theta-(BEDT-TTF)2MZn(SCN)4 (M = Cs, Rb) follow the power law with a large exponent (e.g., 8.4 at 0.29 K for M = Cs) over a wide range of currents in the low-temperature insulating state. The power-law characteristics are attributed to electric field-induced unbinding of electron-hole pairs that are thermally excited in the background of the two-dimensional charge order. The magnitude of crossover electric fields from Ohmic to the power-law characteristics indicates that the electron-electron Coulomb interaction is significantly long-ranged: The screening length is greater than 10 molecule sites.