Thickness accommodation is one of the most significant challenges in creating an origami flasher array. If it is a membrane array, thickness accommodation is generally simpler as panel deformation allows the array to wrap about itself with relative ease. However, for rigid-panel arrays, thickness accommodation is considerably more complicated. This work develops thickness accommodation techniques that yield origami-inspired mechanisms of constant thickness and high-volume efficiency. In this paper, current methods for accommodating thickness in flashers are presented and issues associated with those methods are discussed. Two methods for accommodating thickness in flashers such that panels are constant thickness are proposed.