This article is concerned with decentralised output regulation of hierarchical systems subject to input and output disturbances. It is assumed that the disturbance can be represented as the output of an autonomous linear time invariant (LTI) system with an unknown initial state. The primary objective is to design a decentralised controller with the property that not only does it reject the degrading effect of the disturbance on the output (to achieve a satisfactory steady-state performance), it also results in a small linear quadratic (LQ) cost function (implying a good transient behaviour). To this end, the underlying problem is treated in two phases. In the first step, a number of modified systems are defined in terms of the original system. The problem of designing an LQ centralised controller which stabilises all the modified systems and rejects the disturbance in the original system is considered, and it is shown that this centralised controller can be found efficiently by solving a linear matrix inequality (LMI) problem. In the second step, a method recently presented in the literature is exploited to decentralise the designed centralised controller. It is shown that the obtained controller satisfies the prescribed design specifications including disturbance rejection. Finally, in a more pragmatic context, the system is assumed to be subject to input delay, and a robustness analysis is carried out accordingly. Simulation results elucidate the efficacy of the proposed control law.