We investigate the nature of a sample of 92 Spitzer MIPS 24 km-selected galaxies in the CDF-S, showing power-law-like emission in the Spitzer IRAC 3.6-8 km bands. The main goal is to determine whether the galaxies not detected in X-rays (47% of the sample) are part of the hypothetical population of obscured AGNs not detected even in deep X-ray surveys. The majority of the IR power-law galaxies are ULIRGs at z > 1, and those with LIRG-like IR luminosities are usually detected in X-rays. The optical-to-IR SEDs of the X-ray-detected galaxies are almost equally divided between a BLAGN SED class (similar to an optically selected QSO) and an NLAGN SED (similar to the BLAGN SED but with an obscured UV/optical continuum). A small fraction of SEDs resemble warm ULIRGs (e.g., Mrk 231). Most galaxies not detected in X-rays have SEDs in the NLAGN+ULIRG class as they tend to be optically fainter and possibly more obscured. Moreover, the IR power-law galaxies have SEDs significantly different from those of high-z (z sub(sp) > 1) IR (24 km) selected and optically bright (VVDS I sub(AB) , 24) star-forming galaxies whose SEDs show a very prominent stellar bump at 1.6 km. The galaxies detected in X-rays have 2-8 keV rest-frame luminosities typical of AGNs. The galaxies not detected in X-rays have global X-ray-to-mid-IR SED properties that make them good candidates to contain IR-bright X-ray-absorbed AGNs. If all these sources are actually obscured AGNs, we would observe a ratio of obscured to unobscured 24 km-detected AGNs of 2:1, whereas models predict a ratio of up to 3:1. Additional studies using Spitzer to detect X-ray-quiet AGNs are likely to find more such obscured sources.