This paper studies structured systems, namely linear systems where the state-space matrices have zeros in some fixed positions and free parameters in all other entries. This paper focuses on time-invariant systems in discrete time affected by an unknown input, and their delay-L left invertibility, namely the possibility to reconstruct the input sequence from the output sequence, assuming that the initial state is known, and requiring that the inputs can be reconstructed up to L time steps before the current output. Building upon classical results on linear systems theory and on structured systems, a graphical characterization is obtained of the integers L for which a structured system is generically delay-L left invertible.