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The microbunching instability developed during the beam compression process in the linear accelerator (LINAC) of a free-electron laser (FEL) facility has always been a problem that degrades the lasing performance, and even no FEL is able to be produced if the beam quality is destroyed too much by the instability. A common way to suppress the microbunching instability is to introduce extra uncorrelated energy spread by the laser heater that heats the beam through the interaction between the electron and laser beam, as what has been successfully implemented in the Linac Coherent Light Source and Fermi@Elettra. In this paper, a simple and effective scheme is proposed to suppress the microbunching instability by adding two transverse gradient undulators (TGU) before and after the magnetic bunch compressor. The additional uncorrelated energy spread and the density mixing from the transverse spread brought up by the first TGU results in significant suppression of the instability. Meanwhile, the extra slice energy spread and the transverse emittance can also be effectively recovered by the second TGU. The magnitude of the suppression can be easily controlled by varying the strength of the magnetic fields of the TGUs. Theoretical analysis and numerical simulations demonstrate the capability of the proposed technique in the LINAC of an x-ray free-electron laser facility.