Estimates of the time-to-melt for cryogenic DT inertial fusion targets in the presence of thermal radiation are presented. This time is defined as that required for thermal radiation in a hypothetical reactor to raise the temperature of small polymer capsules containing solid DT by 1 K and to fully liquefy the contents. The time estimates are in turn based on estimates of the infrared absorption spectra of both solid DT and the polymer capsule material. Assuming typical target dimensions and rapid equilibration of the target temperature, the estimates show that the absorption of thermal radiation and subsequent heating of likely capsule materials will dominate the corresponding quantities of DT ice and thus that the former effect largely determines the time-to-melt of the target. Specific estimates are made for capsules fabricated from KaptonTM polyimide. Comparisons are also made for capsules coated with reflective metal coatings, and the potential benefit of these coatings is discussed.