A method of image reconstruction from scattering measurements for use in ultrasonic imaging is presented. The method employs distorted-wave Born iteration but does not require using a forward-problem solver or solving large systems of equations. These calculations are avoided by limiting intermediate estimates of medium variations to smooth functions in which the propagated fields can be approximated by phase perturbations derived from variations in a geometric path along rays. The reconstruction itself is formed by a modification of the filtered-backpropagation formula that includes correction terms to account for propagation through an estimated background. Numerical studies that validate the method for parameter ranges of interest in medical applications are presented. The efficiency of this method offers the possibility of real-time imaging from scattering measurements.