I'm assuming someone must have scooped me on this simple argument. Where does it (first) appear in the literature?

Fix an ultrafilter $\mu$ on $\omega$, the natural numbers.

Alice and Bob play a nim-like game. At the start each player "holds" the empty set and the the starting "position" consists of $\omega$. Beginning with Alice, each player in turn will remove a non-empty finite initial segment from the current position (leaving some final segment of $\omega$) and deposit the removed segment into his or her holdings. Play proceeds for $\omega$ rounds.

Now the object of the game: to finish with holdings that belong to the ultrafilter $\mu$.

Strategy stealing obviates the possibility of either player possessing a winning strategy; the existence of $\mu$ thus contradicts AD. In more detail, if either player has a winning strategy, the game must admit infinitely many winning positions, but that would allow the other player to possibility of moving to a winning position on his or her very first move.

Many sources work much harder than this to prove the weaker results that AC contradicts AD. (Afterthought: I'd love to see a big-list question collecting theorems where unnecessarily complicated proofs permeate the literature despite the availability of simpler treatments...MO appropriate?)