This paper addresses the problem of “ℋ∞ almost synchronization” for networks of identical, linear agents under directed communication topologies. Agents are presumed to be non-introspective; i.e. agents are not aware of their own state or output, and every agent is only provided with a linear combination of its own output relative to that of the neighbors. Providing the solvability 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∞ norm of the corresponding closed-loop transfer function, is reduced to any arbitrarily small value. Thus, synchronization with any desired accuracy can be achieved.