FlyerTalker mspel found great fares on LAN to Iguazu Falls.
The fare is valid from Los Angeles, New York, Orlando and Miami. Orlando starts at an awesome $272 and Los Angeles and New York are not far behind.
You can find dates using ITA Matrix and book on Priceline. Priceline.com provides free 24-hour cancellation.
The flights are on LAN and they supposedly earn 100% miles with AAdvantage if the fare classes actually match.
From IGU, it is a short trip to Buenos Aires (using Avios or cash) or from IGU you can reach several Brazilian cities using TAM. Keep in mind that Argentina and Brazil each charge a reciprocity fee for US passport holders.