/actions: Flux action creators
/components: These are all the JSX React components (the presentation layer) - so there are subfolders like "pages", "controls", etc.
/reducers: These are Redux reducers - they're the logic that handles all of the application state
/services: This is what your question was really about. We use basic ES6 modules for services, with methods like "fetchUser(id)" which use the new JS Fetch API behind the scenes. We then have redux-thunk and redux-promise middleware so we can easily offload state manipulation to these services without actually putting state logic in them.
I've set up a few React projects at this point and this always seems to keep a nice separation of logic. I'm never asking where data is coming from or where it's going. I think Redux has been the biggest difference maker when dealing with all of these parts.
It's no framework, but it's composed of well understood libraries, each with their own important function. You build the structure out yourself so you know exactly how it works, and the pattern will be very familiar to anyone who has used React before.