The concept of space mapping is applied, for the first time, to the design of microwave dispersive delay structures (DDSs). DDSs are components providing specified group delay versus frequency responses for real-time radio systems. The DDSs considered in this paper are formed by cascaded coupled C-sections. It is first shown that aggressive space mapping does not provide sufficient accuracy in the synthesis of DDSs. To address this issue, we propose a predistortion space mapping technique. Compared to aggressive space mapping, this technique provides enhanced accuracy, while compared to output space mapping, it provides greater implementation simplicity. Two full-wave and one experimental examples are provided to illustrate the proposed predistortion space mapping technique.