Whilst rigid, planar surfaces are often used to study cell migration, a physiological scenario requires three-dimensional (3D) scaffolds with tissue-like stiffness. This paper presents a method for fabricating periodic hydrogel scaffolds with a 3D honeycomb-like structure from colloidal crystal templates. The scaffolds, made of hydrogel-walled cavities interconnected by pores, have separately tuneable internal dimensions and adjustable gel stiffness down to that of soft tissues. In conjunction with confocal microscopy, these scaffolds were used to study the importance of cell compliance on invasive potential. Acute promyelocytic leukaemia (APL) cells were differentiated with all-trans retinoic acid (ATRA) and treated with paclitaxel. Their migration ability into the scaffolds' size-restricted pores, enabled by cell softening during ATRA differentiation, was significantly reduced by paclitaxel treatment, which interferes with cell shape recovery. These findings demonstrate the usability of the scaffolds for investigating factors that affect cell migration, and potentially other cell functions, in a realistic 3D tissue model.