Update 2. I've created mdpreview project - self-made solution to preview markdown files in browser. The code is written in Rust. Short introduction is available here.
Update 1. For reStructuredText there is an excellent restview library which works out-of-the-box.
It took half-a-workday time to find suitable solution to be able to somehow preview rendered markdown files.
GitHub uses GitHub Flavored Markdown so before committing anything to the Jukoro reporsitory it would be great to take a quick look at rendered markdown files.
My editor of choice is vim and I have rather strong intension to keep it as is.
I prefer simple solutions meaning right after save I can see changes rendered in browser (I don't need really live preview to see changes while typing). Browser-based editors don't suit me due to the fact you have to edit a file in browser.
I looked at Atom and almost decided to use it but a small annoyance forced me to look better for other ways (the annoyance is about caret disappearing in editor after switching from and back to it). For me there is one more problem with it. I'd like to turn off bold font in editor (the way I get used to working in terminal) but I have no luck searching for this feature in preferences.
There are several nodejs-based solutions but with most of them I had a problem with live update (for some reason it was not working as expected by me).
markdown-live is the solution that I chose. It works great and have a side panel to quickly switch between markdown-files. Just great.