We elucidate a significant thermal rectification effect in molecule bridges (MBs) that covalently bond gold and carbon nanotubes (CNTs) by non-equilibrium molecular dynamics (MD) simulations. Remarkably, we find asymmetric heat flux across these MBs, resulting in significant thermal rectification. Thermal rectification ratios are found to decrease with the number of the MBs and increase with the lengths of CNTs, reaching a value of as high as 3.75. In addition, the calculations show that the thermal conductances vary with orientation of the crystal planes of Au, the number of molecular bridges, and the length of the CNTs. They are in a range of 40-250 MW m-2 K-1 , agreeing well with the reported experimental results. Detailed analysis suggests that the thermal rectification can be attributed to the mismatch of vibrational modes between neighboring sulfur and carbon atoms in the MBs. This discovery could provide theoretical guidance in designing molecular thermal diodes, paving new routes to potential applications in nanoscale heat manipulation, waste energy harvest, chip cooling, and molecular phononics.