Azure web sites can be conveniently deployed by pushing their git repository to a deployment URL.
Is the same thing possible with worker roles? I haven't found any information about this on the web.
This functionality is not natively available (i.e. built-in) to cloud services (Web/Worker roles). However you do have full flexibility to implement your own solution which pulls contents from GitHub/wherever, and there are some projects that seek to provide this functionality, if that helps.