You can format your document using text shortcuts instead of keyboard ones with the new Markdown support for the web. In a post on its website, the company says that it will format the text for you after you type it. It will automatically get converted to a level one heading if you type.
There are a few autocorrections for bulleted and numbered lists in the doc. You can now use Markdown to add headings, bold and italicize text (or do both), strikethrough, though it is done using either side of your content. It's not a complete implementation of Markdown, but at least it covers most of what I use the language for.
The feature can be activated by going to Tools and Preferences and checking the box. If you don't see it, it may not be rolled out for your account yet.
If you're used to writing in markdown in other applications, implementation is probably going to take some getting used to. Rather than showing you your marks in plain text, it uses them to apply formatting and get rid of them. That is different from how most other text editors display Markdown, with the editor adding some type of formatting to give you an idea of how it works.
It's probably a personal preference whether you like this approach or not. The people who use Markdown to get complete control of their text probably won't be interested in the implementation. For anyone who just wants the ability to use Markdown as a shortcut to formatting, and doesn't care about fussing around with plaintext, it could be relatively easy to use.
It's worth noting that this implementation is much friendlier for when you share a document with a coworker who doesn't know what Markdown is.
It's easy to imagine a lot of people getting confused if typing a pound sign in front of something, and that's why the feature is off by default. I think it's time to start a campaign to get Google to add a Vim mode to Docs since it's getting into the habit of adding fun nerdy features.