Although Head Injury Criterion (HIC) is an effective criterion for head injuries caused by linear acceleration such as skull fractures, no criteria for head injuries caused by rotational kinematics has been accepted as effective so far. This study proposed two criteria based on angular accelerations for Traumatic Brain Injury (TBI), which we call Rotational Injury Criterion (RIC) and Power Rotational Head Injury Criterion (PRHIC). Concussive and non-concussive head acceleration data obtained from football head impacts were utilized to develop new injury criteria. A well-validated human brain Finite Element (FE) model was employed to find out effective injury criteria for TBI. Correlation analyses were performed between the proposed criteria and FE-based brain injury predictors such as Cumulative Strain Damage Measure (CSDM), which is defined as the percent volume of the brain that exceeds a specified first principal strain threshold, proposed to predict Diffuse Axonal Injury (DAI) which is one of TBI. The RIC was significantly correlated with the CSDMs with the strain thresholds of less than 15% (R > 0.89), which might predict mild TBI. In addition, PRHIC was also strongly correlated with the CSDMs with the strain thresholds equal to or greater than 20% (R > 0.90), which might predict more severe TBI.