Fe–Cr alloys are widely used in power, petroleum, and manufacturing industries for their good resistance to corrosion and oxidation at high temperatures. Ultrasound is the only nondestructive method so far to measure the residual stress of in-service components. However, for parts with material anisotropy, such as materials processed by rolling, the measurement accuracy is highly restrained. In this paper, a rolled 316L stainless steel sample is used to study the influence of texture on the measurement of residual stress by ultrasonic surface wave. The experimental results show that the propagation velocity of surface waves in the sample has anisotropic characteristics. The wave velocity parallel to the rolling direction (0°) is the maximum, and the wave velocity perpendicular to the rolling direction (90°) is the minimum, thereby affecting the measurement accuracy. It is found that reducing the frequency of surface waves can reduce the influence of anisotropy. Therefore, a low-frequency method and modified formula are used to improve the measurement accuracy. The maximum error in the rolling direction is reduced from 21.3 to 3.6 MPa, and the maximum relative error is also reduced from 45.4 to 9.0%. The modified formula can further reduce the influence of anisotropy, with the maximum error value further reduced to 2.3 MPa, the maximum relative error reduced to 4.9%, and the surface wave detection accuracy effectively improved.