Spitzer spectroscopy has revealed that 80% of submm galaxies (SMGs) are starburst (SB)-dominated in the mid-infrared. Here we focus on the remaining 20% that show signs of harboring powerful active galactic nuclei (AGNs). We have obtained Spitzer-InfraRed Spectrograph spectroscopy of a sample of eight SMGs that are candidates for harboring powerful AGNs on the basis of IRAC color selection (S 8 μm/S 4.5 μm>2, i.e., likely power-law mid-infrared spectral energy distributions). SMGs with an AGN dominating (50%) their mid-infrared emission could represent the "missing link" sources in an evolutionary sequence involving a major merger. First of all, we detect polycyclic aromatic hydrocarbon (PAH) features in all of the SMGs, indicating redshifts from 2.5 to 3.4, demonstrating the power of the mid-infrared to determine redshifts for these optically faint dusty galaxies. Second, we see signs of both star formation (from the PAH features) and AGN activity (from continuum emission) in our sample: 62% of the sample are AGN-dominated in the mid-infrared with a median AGN content of 56%, compared with <30% on average for typical SMGs, revealing that our IRAC color selection has successfully singled out sources with proportionately more AGN emission than typical SB-dominated SMGs. However, we find that only about 10% of these AGNs dominate the bolometric emission of the SMG when the results are extrapolated to longer infrared wavelengths, implying that AGNs are not a significant power source to the SMG population overall, even when there is evidence in the mid-infrared for substantial AGN activity. When existing samples of mid-infrared AGN-dominated SMGs are considered, we find that S 8 μm/S 4.5 μm>1.65 works well at selecting mid-infrared energetically dominant AGNs in SMGs, implying a duty cycle of ~15% if all SMGs go through a subsequent mid-infrared AGN-dominated phase in the proposed evolutionary sequence.