This paper contributes to the research direction of utilizing pre-trained generative diffusion models as prior probabilities for solving Bayesian inverse problems. We design a sequential Monte Carlo method for the linear-Gaussian inverse problem, which is based on "decoupled diffusion", where the generation process is designed to allow larger updates for samples. The method is asymptotically accurate, and we demonstrate the effectiveness of the Decoupled Diffusion Sequential Monte Carlo (DDSMC) algorithm on synthetic data as well as protein and image data. We also show how to extend this approach to discrete data.