A large-scale off-lattice Monte Carlo simulation of main-chain liquid crystal elastomers (LCE) is performed to explore the possibility of finding temperature-induced biaxial orientational ordering in such systems. The liquid-crystalline molecules of the LCE are modeled by biaxial soft-core Gay-Berne ellipsoids and embedded into a monodomain elastomer network linked together by finitely extendable nonlinear elastic (FENE) bonds. On cooling from the uniaxial nematic phase, biaxial ordering of ellipsoids is observed and is accompanied by a spontaneous biaxial elastic deformation of the sample. Biaxial orientational alignment is also reflected in the calculated deuterium magnetic resonance spectra and in the predicted scattered X-ray intensity patterns, while no major calorimetric anomaly is observed at the uniaxial-to-biaxial nematic phase transition. The long-range character of the observed biaxial phase is elucidated by calculating selected orientational correlation functions.