What worked for me was simply using the app name as root:

import myMP3File from '<appname>/assets/mymp3.mp3';
const Sound = require('react-native-sound');
Sound.setCategory('Playback');

// Do whatever you like with it.
Sound(myMP3File, () => console.log('soundfile loaded!'));

Edit:

We now use rn-fetch-blob and the following solution to access local files:

import RNFetchBlob from 'rn-fetch-blob';
const { fs } = RNFetchBlob;
filePathIos = `${fs.dirs.MainBundleDir}/yourFolder/yourMp3.mp3`;
filePathAndroid = fs.asset('yourFolder/yourMp3.mp3');

The corresponding path can then be used to copy the file using fs.cp().

This will preload the sound and when you press the play button it will play it.

export default class MovieList extends Component {
    componentDidMount(){
      this.hello = new Sound('whoosh.mp3', Sound.MAIN_BUNDLE, (error) => {
        if (error) {
          console.log('failed to load the sound', error);
          return;
        }
      });
    }
    
    
    handlePress() {
      this.hello.play((success) => {
        if (!success) {
          console.log('Sound did not play')
        }
      })
    }

    render() {
        const { movie } = this.props
        return (
            <TouchableOpacity onPress={this.handlePress.bind(this)}>
                <View>
                      <Text>Start</Text>
                </View>
            </TouchableOpacity>
        )
    }
}

If you are looking to play sound tracks from a list of sounds please check this gist for detailed code.