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Polaritonic materials, owing to a strong phonon-polariton resonance in the THz and far-infrared parts of the electromagnetic spectrum, offer both high-index dielectric and metallic response in this regime. This complex response makes them suitable candidates for the design of metamaterial-related phenomena and applications. Here we show that one type of polaritonic-material-based structures that are particularly suitable for the achievement of a wide range of metamaterial properties are systems of polaritonic rods. To study the interplay between the material and the structural resonances in such systems we employ as model systems rods of LiF and SiC and we calculate first the scattering properties of a single rod, identifying and discussing the behavior of the different resonances for different rod diameters. To analyze the response of ensembles of polaritonic rods we employ an effective medium approach based on the Coherent Potential Approximation (CPA), which is shown to be superior to the simple Maxwell-Garnett approximation for polaritonic and high-index dielectric metamaterials. Calculating and analyzing the CPA effective parameters, we found that our systems exhibit a large variety of interesting metamaterial properties, including hyperbolic dispersion, epsilon-near-zero and negative refractive index response. This rich response, achievable in almost any system of polaritonic rods, is highly engineerable by properly selecting the radius and the filling ratio of the rods, making polaritonic rod systems an ideal platform for demonstration of multifunctional metamaterials.