We propose an innovative approach for the realization of a microwave absorber fully transparent in the optical regime. This device is based on the Salisbury screen configuration, which consists of a lossless spacer, sandwiched between two graphene sheets whose sheet resistances are different and properly engineered. Experimental results show that it is possible to achieve near-perfect electromagnetic absorption in the microwave X-band. These findings are fully supported by an analytical approach based on an equivalent circuital model. Engineering and integration of graphene sheets could facilitate the realization of innovative microwave absorbers with additional electromagnetic and optical functionalities that could circumvent some of the major limitations of opaque microwave absorbers.