This paper describes a method for systematically obtaining distributions of annealing and/or creation energies, tunneling distances, and trap depths for specific defect populations from electron-spin-resonance (ESR) transients. The method of analysis, specifically adapted to the constraints imposed by ESR, is presented for obtaining defect distributions from a few transients using time-dependent temperature variations. Specific important cases, numerical examples, and application to actual data are examined. A brief discussion of the experimental methods required to obtain the ESR transients with sufficient sensitivity are described. Finally, issues of resolution and noise, comparison to other transient analysis methods, and possible extensions are presented. A detailed discussion of the application of the method to hydrogenated amorphous silicon is presented in the following paper.