A hexagonal graphene-like boron-carbon-nitrogen (h-BCN) monolayer, a new two-dimensional (2D) material, has been synthesized recently. Herein we investigate for the first time the thermal conductivity of this novel 2D material. Using molecular dynamics simulations based on the optimized Tersoff potential, we found that the h-BCN monolayers are isotropic in the basal plane with close thermal conductivity magnitudes. Though h-BCN has the same hexagonal lattice as graphene and hexagonal boron nitride (h-BN), it exhibits a much lower thermal conductivity than the latter two materials. In addition, the thermal conductivity of h-BCN monolayers is found to be size-dependent but less temperature-dependent. Modulation of the thermal conductivity of h-BCN monolayers can also be realized by strain engineering. Compressive strain leads to a monotonic decrease in the thermal conductivity while the tensile strain induces an up-then-down trend in the thermal conductivity. Surprisingly, the small tensile strain can facilitate the heat transport of the h-BCN monolayers.