We propose a plasmonic structure to obtain polarization-insensitive localized surface plasmon resonance (LSPR) sensor, which consists of cross-slit metallic periodic arrays embedded in the background material. Numerical simulation illustrates that the mechanism of the LSPR sensor is based on the shift of the Fabry–Perot cavity mode resonance peak in the spectrum as the change of the dielectric material properties for the near fields. And one of the transmission dips of the structure is very sensitive to the background materials; the structure could gain the sensitivity (nm/RIU) more than 500nm/RIU. Meanwhile, the structure holds great potential to achieve high-performance sensors in practical application due to polarization-insensitive virtue.