We examine the 24 μm to X-ray color of 157 X-ray-selected active galactic nuclei (AGNs) as a function of X-ray obscuration and optical classification in the Chandra Deep Field-South. The sample consists of the Chandra hard-band detections with 2-8 keV flux above 10-15 ergs s-1 cm-2. A deep 24 μm mosaic obtained with Spitzer provides mid-infrared fluxes for the sample. Since obscured AGNs locally have higher 24 μm/2-8 keV flux ratios than unobscured AGNs, and since X-ray background models predict a large population of obscured AGNs, we expect to find many X-ray-hard, IR-bright AGNs. Instead, we find that the 24 μm to X-ray flux ratio does not depend on X-ray hardness in the full sample, nor does it differ between narrow- and broad-line AGNs. We identify five nearly Compton-thick AGNs and find they have similar 24 μm to X-ray flux ratios compared to the full sample. We consider AGNs in the narrow redshift spikes at z ~ 0.7; for these AGNs, there is some evidence that the flux ratio increases with X-ray hardness. The redshift slice also shows an odd trend that is also prominent in the full sample: a group of X-ray-hard AGNs with very low 24 μm to X-ray flux ratios.