About the cheapest I've found is from LA to Santo Domingo, $550 roundtrip from Priceline.
Are there any smaller, 'specialty' airlines offering trips to Santo Domingo for cheap?
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For random dates in February-March, the cheapest LAX-SDQ return trip I could quickly find was about $376, on Spirit Airlines.
Dunno about "specialty" airlines, but if you're flexible with dates, and don't mind long connection times, you can get quite cheap flights. Try e.g. skyscanner.com and the "whole month" and "whole year" options.