Graphene-like nanomembranes made of the neighboring elements of boron, carbon and nitrogen elements, are well-known of showing outstanding physical properties. Herein, with the aid of density functional theory (DFT) calculations, various atomic configurations of the graphene-like BCN nanosheets are investigated. DFT results reveal that depending on the atomic arrangement, the BCN monolayers may display semimetallic Dirac cone or semiconducting electronic nature. BCN nanosheets are also found to exhibit high piezoelectricity and carrier mobilities with considerable in-plane anisotropy, depending on the atomic arrangement. For the predicted most stable BCN monolayer, thermal and mechanical properties are explored using machine learning interatomic potentials. The room temperature tensile strength and lattice thermal conductivity of the most stable BCN monolayer are estimated to be orientation-dependent and remarkably high, over 78 GPa and 290 W/m.K, respectively. In addition, the thermal expansion coefficient of the monolayer BCN at room temperature is estimated to be −3.2 × 10−6 K−1, which is close to that of the graphene. The piezoelectric response of the herein proposed BCN lattice is also predicted to be close to that of the h-BN monolayer. Presented results highlight outstanding physics of the BCN nanosheets.