This is one of those rare occasions when you can use a guitar pick without me telling you off. There are many notes doubled up in this riff. One option is to make ‘running man’ shapes with your index and middle fingers. However, if you want to keep the muted the strings (by resting the edge of your picking hand lightly on the strings just above the bridge) your only option is to use a pick.
If you decide against the pick, you might prefer to play it this way:
Requested by Ray