A robust, novel, to the best of our knowledge, fiber Mach-Zehnder interferometer strain sensor is designed and experimentally implemented. The sensor consists of two concave-lens-like long-period fiber gratings and is engraved by the high frequency ${{\rm CO}_2}$CO2 laser. The concave-lens-like grids can excite higher-order cladding modes to interfere with the fundamental mode, which increases the light-material contact. The excellent interference spectra are obtained and analyzed theoretically. The experimental results show that the strain sensitivity of the sensor can reach 0.011 dB/µ$\unicode{x03B5}$ε in the range of 0-2160 µ$\unicode{x03B5}$ε. The resolution of the sensor is up to 0.91 µ$\unicode{x03B5}$ε. Moreover, the temperature crosstalk can be self-compensated by monitoring a pair of split interfering dips. These outstanding characteristics make it very suitable as a candidate for strain measurement.