This paper addresses the problem of "H-infinity, almost synchronization" for homogeneous networks of general linear agents subject to external disturbances and under directional communication links. Agents are presumed to be non-introspective; i.e. agents are not aware of their own states or outputs, and the only available information for each agent is a network measurement that is a linear combination of relative outputs. Under a certain set of conditions, a family of dynamic protocols is developed such that the impact of disturbances on the synchronization error dynamics, expressed in terms of the H-infinity norm of the corresponding closed-loop transfer function, is reduced to any arbitrarily small value. (C) 2014 Elsevier Ltd. All rights reserved.