A THz-to-IR converter can be an effective solution for the detection of low-IR-signature targets by combining the advantages of mature IR detection mechanisms with high atmospheric transmittance in the THz region. A metallic metasurface (MS)-based absorber with linear polarization dependence based on a split-ring resonator (SRR) unit cell has been previously studied as a preliminary example of a THz-to-IR converter structure in the literature. In this simulation-based study, a new cross-shaped unit cell-based metallic MS absorber structure sensitive to dual polarization is designed to eliminate linear polarization dependency, thereby allowing for incoherent detection of THz radiation. A model is developed to calculate the temperature difference and the response time for this new cross-shaped absorber structure, and its performance is compared to the SRR structure for both coherent and incoherent illumination. This model allows for understanding the efficiency of these structures by considering all loss mechanisms which previously had not been considered. It is found that both structures show similar performance under linearly polarized coherent illumination. However, under incoherent illumination, the IR emittance efficiency as gauged by the temperature difference for the cross-shaped structure is found to be twice as high as compared to the SRR structure. The results also imply that calculated temperature differences for both structures under coherent and incoherent illumination are well above the limit of the minimum resolvable temperature difference of the state-of-the-art IR cameras. Therefore, dual-polarized or multi-polarization-sensitive MS absorber structures can be crucial for developing cost-effective THz-to-IR converters and be implemented in THz imaging solutions.