My talk at React.js Conf is on exactly this topic: how to integrate React and TypeScript. For JSX the approach I recommend is a bit of a hack but works quite well. You basically tag the JSX in such a way that the IDE is happy. Then in a precompile step you find, transform, and replace the JSX[1] prior to passing to TypeScript for final compilation. You lose out on intellisense and autocompletion but you still get type-checking and can still use some ES6 features in your JSX like fat arrows.

1. https://github.com/jbrantly/ts-jsx-loader (for webpack, but basic approach works in gulp/grunt as well)