We present Spitzer-IRAC and MIPS mid-IR observations of a sample of 76 radio-detected submillimeter-selected galaxies (SMGs) with spectroscopic redshifts, the largest such sample published to date. From our data, we find that IRAC colors of SMGs are much more uniform as compared with rest-frame UV and optical colors, and z>1.5 SMGs tend to be redder in their mid-IR colors than both field galaxies and lower-z SMGs. However, the IRAC colors of the SMGs overlap those of field galaxies sufficiently that color-magnitude and color-color selection criteria suggested in the literature to identify SMG counterparts produce ambiguous counterparts within an 8" radius in 20%-35% of cases. We use a rest-frame J-H versus H-K color-color diagram and a S24/S8.0 versus S8.0/S4.5 color-color diagram to determine that 13%-19% of our sample are likely to contain active galactic nuclei which dominate their mid-IR emission.