There's nothing special from the RMarkdown end of things that makes the rendering happen. GitHub renders markdown (it's even nicer in .md format, which is why you'll see a lot of README.Rmd rendered to README.md) with "GitHub flavored markdown" (you'll see it abbreviated as gfm).
I don't use BitBucket, but it looks like it supports markdown rendering. One option would be to output to .md. For example, if I'm setting up the YAML for an html file, but want to keep the markdown, I'd do this (see the R Markdown "Keeping Markdown" section for details):
---
title: "My document"
output:
html_document:
keep_md: true
---
Note that YAML is whitespace sensitive, so make sure to match the indents correctly!
Hope this helps.