Living tissues are non-linearly elastic materials that exhibit viscoelasticity and plasticity. Man-made, implantable bioelectronics mainly rely on rigid or elastic encapsulation materials and brittle thin films of metal that can be manipulated with microscopic precision to offer reliable electrical properties. Here, we engineer a surface microelectrode array that replaces both the traditional encapsulation and conductive components with viscoelastic materials. Our entirely viscoelastic array overcomes previous limitations in matching the stiffness and relaxation behavior of soft biological tissues by using hydrogels as the outer layers. We introduce a novel hydrogel-based conductor made from an ionically conductive alginate matrix enhanced with carbon nanomaterials (graphene, carbon nanotubes). These high aspect ratio additives provide electrical percolation even at low loading fractions, and we fabricate ultra-soft viscoelastic conductive electrodes and electrical tracks that intimately conform to the convoluted surface of the heart or the brain cortex. Our combination of conducting and insulating viscoelastic materials, with top-down manufacturing, allows for the versatile fabrication of electrode arrays compatible with standard electrophysiology platforms, and offer promising applications in bioengineering and nanomedicine for recording and stimulation.