In this study, the thermal transport properties of hexagonal monolayer BN, AlN, and GaN nanoribbons are systematically investigated using non-equilibrium molecular dynamics simulations. Firstly, the effect of length, width, and temperature on the lattice thermal conductivities of these ribbons are explored. It is observed that increases in sample geometry lengths of simulations and decreases in temperature lead to an increase in the thermal conductivity of all three materials. Secondly, thermal rectification coefficients of ribbons are calculated by creating asymmetric thermal contacts through non-equilibrium molecular dynamics simulations. Thus, it is shown that thermal rectification factors of up to 20% can be achieved by manipulating the thermal asymmetry ratio in pristine structures without asymmetry of group-III nitride nanoribbons. These results imply that group-III nitride nanoribbons may be considered as an alternative option to control the heat flow in thermal management and thermoelectric device applications.