We report the discovery of blueshifted (Δv>200 km s-1) mid-infrared [Ne III] and/or [Ne V] emission in 25 out of 82 ultraluminous infrared galaxies (ULIRGs; 30% of our sample). The incidence of blueshifted [Ne V] emission is even higher (59%) among the sources with a [Ne V] detection - the tell-tale signature of an active galactic nucleus (AGN). Sixteen ULIRGs in our sample, 11 of which are optically classified as AGNs, have [Ne III] blueshifts above 200 km s-1. A comparison of the line profiles of their 12.81 μm [Ne II], 15.56 μm [Ne III], and 14.32 μm [Ne V] lines reveals the ionization of the blueshifted gas to increase with blueshift, implying decelerating outflows in a stratified medium, photoionized by the AGN. The strong correlation of the line width of the [Ne III] line with the radio luminosity indicates that the interaction of expanding radio jets with the dense interstellar medium surrounding the AGN may explain the observed neon line kinematics for the strongest radio sources in this sample.