Radio-frequency (RF) waveform synthesis has broad applications in ultrawide-bandwidth wireless communications, radar systems, and electronic testing. Photonic-based approaches offer key advantages in bandwidth and phase noise thanks to the ultrahigh optical carrier frequency. In this work, we demonstrate Fourier synthesis arbitrary waveform generation (AWG) with integrated optical microresonator solitons. The RF temporal waveform is synthesized through line-by-line amplitude and phase shaping of an optical soliton microcomb, which is down-converted to the RF domain through dual-comb optical coherent sampling. A variety of RF waveforms with tunable repetition cycles are shown in our demonstration. Our approach provides not only the possibility of precise Fourier synthesis at microwave and millimeter-wave frequencies, but also a viable path to fully integrated photonic-based RF AWG on a chip.