Tensile testing is an essential experiment to assess the mechanical integrity of musculoskeletal soft tissues, yet standard test methods have not been developed to ensure the quality and reproducibility of these experiments. The ASTM International standards organization has created tensile test standards for common industry materials that specify geometric dimensions of test specimens (coupons) that promote valid failures within the gage section (midsubstance), away from the grips. This study examined whether ASTM test standards for plastics, elastomers, and fiber-reinforced composites are suitable for tensile testing of bovine meniscus along the circumferential fiber direction. We found that dumbbell (DB) shaped coupons based on ASTM standards for elastomers and plastics had an 80% and 60% rate of midsubstance failures, respectively. The rate of midsubstance failures dropped to 20% when using straight (ST) coupons based on ASTM standards for fiber-reinforced composites. The mechanical properties of dumbbell shaped coupons were also significantly greater than straight coupons. Finite element models of the test coupons revealed stress distributions that supported our experimental findings. In addition, we found that a commercial deli-slicer was able to slice meniscus to uniform layer thicknesses that were within ASTM dimensional tolerances. This study provides methods, recommendations, and insights that can advance the standardization of tensile testing in meniscus and other soft fibrous tissues.