We report a fast response microfluidic Fabry-Perot (FP) interferometer refractive index (RI) fiber sensor based on a concave-core photonic crystal fiber (CPCF), which is formed by directly splicing a section CPCF with a section of single mode fiber. The CPCF is made by cleaving a section of multimode photonic crystal fiber with an axial tension. The shallow concave-core of CPCF naturally forms the FP cavity with a very short cavity length. The inherent large air holes in the cladding of CPCF are used as the open channels to let liquid sample come in and out of FP cavity. In order to shorten the liquid channel length and eliminate the harmful reflection from the outside end face of the CPCF, the CPCF is cleaved with a tilted tensile force. Due to the very small cavity capacity, the short length and the large sectional area of the microfluidic channels, the proposed sensor provides an easy-in and easy-out structure for liquids, leading to great decrement of the measuring time. The proposed sensor exhibits fast measuring speed, the measuring time is less than 359 and 23 ms for distilled water and pure ethanol, respectively. We also experimentally study and demonstrate the superior performances of the sensor in terms of high RI sensitivity, good linear response, low temperature cross-sensitivity and easy fabrication.