This project was inspired by https://github.com/devfd/react-native-workers. Since that project has been in a broken state under iOS (I'm not sure about Android), I've decided to write my own.
npm install https://github.com/noitcudni/react-native-background-workers.git --save
rnpm link react-native-background-workers
For release build to work, we'll need to bundle up worker javascript files as well. Your worker files should end with _worker.js and they need to reside in App/Workers relative to your App's root directory.
Shamelessly stolen from devfd/react-native-workers#21. Add the following to your react native app's Build Phases -> Bundle React Native code and images section:
export NODE_BINARY=node
../node_modules/react-native-background-workers/react-native-workers.sh # <-- ADD THIS LINE
../node_modules/react-native/packager/react-native-xcode.sh