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Abstract Recently developed methods for transforming 2D patterns of thin‐film materials into 3D mesostructures create many interesting opportunities in microsystems design. A growing area of interest is in multifunctional thermal, electrical, chemical, and optical interfaces to biological tissues, particularly 3D multicellular, millimeter‐scale constructs, such as spheroids, assembloids, and organoids. Herein, examples of 3D mechanical interfaces are presented, in which thin ribbons of parylene‐C form the basis of transparent, highly compliant frameworks that can be reversibly opened and closed to capture, envelop, and mechanically restrain fragile 3D tissues in a gentle, nondestructive manner, for precise measurements of viscoelastic properties using techniques in nanoindentation. Finite element analysis serves as a design tool to guide selection of geometries and material parameters for shape‐matching 3D architectures tailored to organoids of interest. These computational approaches also quantitate all aspects of deformations during the processes of opening and closing the structures and of forces imparted by them onto the surfaces of enclosed soft tissues. Studies of cerebral organoids by nanoindentation show effective Young's moduli in the range from 1.5 to 2.5 kPa depending on the age of the organoid. This collection of results suggests broad utility of compliant 3D mesostructures in noninvasive mechanical measurements of millimeter‐scale, soft biological tissues.