Having looked into this some more I stumbled upon the document reStructuredText Interpreted Text Roles. From this document:
Interpreted text uses backquotes (`) around the text. An explicit role marker may optionally appear before or after the text, delimited with colons. For example:
This is `interpreted text` using the default role. This is :title:`interpreted text` using an explicit role.
It seems that there is a code role, so you can simply type
:code:`a = b + c`
to render an inline code block. To get syntax highlighting you can define a custom role. For example
.. role:: bash(code)
:language: bash
which you can then use like so:
Here is some awesome bash code :bash:`a = b + c`.
Note that the role definition must be placed before references to the role.
Note, the document I link to makes no mention of the version of docutils to which it refers. The code role is not available in docutils 0.8.1 (which is the only version I have to test against).