A pseudo-spectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms is presented. The method allows the treatment of moderately complex geometries by means of a multi-domain approach and it is able to cope with non-constant fluid properties and non-orthogonal problem domains. In addition, the fully implicit scheme yields improved stability properties as opposed to semi-implicit schemes commonly employed. Key components of the method are a Chebyshev collocation discretization, a special pressure-correction scheme, and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the proposed method is investigated by considering several numerical examples of different complexity, and also includes comparisons to alternative solution approaches based on finite-volume discretizations.