A novel Envelope Following (EF) method is presented for fast and accurate transient simulation of microwave and RF circuits with almost-periodic responses. Targeted circuits contain high-frequency components whose amplitudes vary slowly. The proposed high-order EF method is based on a stable and high-order Obreshkov formula (ObF). Use of high-order formula allows accurate and fast analysis while keeping the same accuracy as the conventional low-order integration methods and is validated by the presented numerical example.