We present a remarkable thermoacoustically driven compression effect based on the conversion of gas flow from an alternating state to a direct state. The alternating gas flow is generated by the thermoacoustic effect in thermoacoustic engines, whereas direct gas flow is achieved by means of the flow rectification effect of check valves. A demonstrative thermoacoustic compressor consisting of two standing-wave thermoacoustic engines, two reservoirs, and three check valves is constructed for experimental investigation. With nitrogen as a working gas and an initial pressure of 2.4 MPa in all components, a usable pressure difference of 0.4 MPa is achieved, with the average gas pumping rate reaching 2.85 Nm3/h during the first 3 s of the compression process. The simple mechanical structure and thermally driven nature of the compressor show potential in gas compression, power generation, and refrigeration applications.