Summary: We consider discontinuous in time and continuous in space Galerkin finite element methods for the numerical solution of reaction-diffusion differential equations. These are implicit methods that require the solution of a system of nonlinear equations at each time node. In this paper, we explore the use of Krylov-subspace techniques for the iterative solution of the linear systems that arise when these nonlinear systems are solved by means of Newton-type methods. It is shown how these linear systems depend on the choice of the basis functions used for the time discretization. We demonstrate that Krylov-subspace methods can be speed up considerably by employing an orthogonal basis for the time discretization and by combining the Krylov iteration with a suitable block preconditioner. Results of numerical experiments are reported.