Multiresonant high impedance surfaces (HIS) without grounding vias that perform as artificial magnetic conductors (AMC) in multiple frequency bands and furthermore exhibit electromagnetic band gaps (EBG) in the same bands are presented. This is achieved by introducing perturbed frequency selective surface (FSS) arrays printed on grounded dielectric substrates. Arrays of linear dipoles are employed as an example. Perturbations are introduced by means of reducing the length of every other array element. Starting from the characteristics of a perturbed free-standing FSS, the authors present the effect of the perturbation on the excited currents and on the reflection properties of a corresponding AMC. Conclusions about the performance limitations are derived. Subsequently, a parametric study on practical HIS is presented and an optimised design with dual-band AMC and EBG response is demonstrated. Method of moments-based software has been developed and utilised for the fast and accurate analysis of such arrays. Experimental results validate the performance of the optimised structure.