Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatistical method for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rather than the conventional least-squares-based approach, to any Gaussian linear inverse problem using a noniterative method. The sequential approach to solving linear and weakly nonlinear problems is computationally efficient compared with traditional least-squares-based inversion. The sequential approach also allows one to solve the inverse problem in only a small part of the model space while conditioned to all available data. From a geostatistical point of view, the method can be used to condition realizations of Gaussian random fields to the possibly noisy linear average observations of the model space.