Optical nanoantennas provide control of light at the nanoscale, which makes them important for diverse areas ranging from photocatalysis and flat metaoptics to sensors and biomolecular tweezing. They have traditionally been limited to metallic and dielectric nanostructures that sustain plasmonic and Mie resonances, respectively. More recently, nanostructures of organic J-aggregate excitonic materials have been proposed capable of also supporting nanooptical resonances, although their advance has been hampered from difficulty in nanostructuring. Here, the authors present the realization of organic J-aggregate excitonic nanostructures, using nanocylinder arrays as model system. Extinction spectra show that they can sustain both plasmon-like resonances and dielectric resonances, owing to the material providing negative and large positive permittivity regions at the different sides of its exciton resonance. Furthermore, it is found that the material is highly anisotropic, leading to hyperbolic and elliptic permittivity regions. Nearfield analysis using optical simulation reveals that the nanostructures therefore support hyperbolic localized surface exciton resonances and elliptic Mie resonances, neither of which has been previously demonstrated for this type of material. The anisotropic nanostructures form a new type of optical nanoantennas, which combined with the presented fabrication process opens up for applications such as fully organic excitonic metasurfaces., (© 2022 The Authors. Advanced Science published by Wiley-VCH GmbH.)