We explore consequences of the Averaged Null Energy Condition (ANEC) for scaling dimensions Delta of operators in four-dimensional N = 1 superconformal field theories. We show that in many cases the ANEC bounds are stronger than the corresponding unitarity bounds on Delta. We analyze in detail chiral operators in the (1/2 j, 0) Lorentz representation and prove that the ANEC implies the lower bound Delta >= 3/2 j, which is stronger than the corresponding unitarity bound for j > 1. We also derive ANEC bounds on (1/2 j, 0) operators obeying other possible shortening conditions, as well as general (1/2 j, 0) operators not obeying any shortening condition. In both cases we find that they are typically stronger than the corresponding unitarity bounds. Finally, we elucidate operator-dimension constraints that follow from our N = 1 results for multiplets of N = 2; 4 superconformal theories in four dimensions. By recasting the ANEC as a convex optimization problem and using standard semidefinite programming methods we are able to improve on previous analyses in the literature pertaining to the nonsupersymmetric case.