What is the best treatment for strained vocal chords?

Rest: As with any muscle strain, rest is the first treatment to try. In this case, no talking. How long will depend on the severity of the strain and whether you're a famous radio personality, opera singer or just a regular joe. A proper ENT doc can help you figure out exactly what is going on and how best to proceed.

Rest & time: If you follow the entertainment industry, especially those who sing, complete voice rest is common for strained vocal cords and even after surgery on said cords. If that doesn't seem to do the trick, go see your ENT for further evaluation. Of course, besides rest & time, other part of solution is to determine cause of strain & address it: http://www.Umm.Edu/altmed/articles/laryngitis-000099.Htm.