A long-period fiber-grating sensor with a high strain sensitivity of $-7.6\mathrm{pm}∕\mathrm{\mu }\epsilon$ and a low temperature sensitivity of $3.91\mathrm{pm}∕°\mathrm{C}$ is fabricated by use of focused ${\mathrm{CO}}_2$ laser beam to carve periodic grooves on a large- mode-area photonic crystal fiber. Such a strain sensor can effectively reduce the cross-sensitivity between strain and temperature, and the temperature-induced strain error obtained is only $0.5\mathrm{\mu }\epsilon ∕°\mathrm{C}$ without using temperature compensation.