I wrote an entire chapter about the topic. There isn't a silver bullet and it's al situational.
On simple projects I stick to basic CSS and then generate a separate CSS to load using the ExtractTextPlugin. Of course this is something that doesn't scale well as you start hitting the problems of CSS, mainly globals.
There are all sorts of ways to deal with that ranging from conventions to pre/postprocessor based solutions. From my point of view CSS Modules seems like the strongest contender at the moment.
Compared to the inline styling based solutions there's less to learn while CSS Modules solve the fundamental problem of globals. It seems something particularly fitting for React given already deal in terms of components. Check out Pattern Lab for a related idea and inspiration.