This was surprisingly underdocumented out in the world. If you've got a Rails controller that ever uses request.body.read, you can set that in a functional test like so:

class ArticlesControllerTest < ActionController::TestCase
  test "creates a new article" do
    attributes = {
      :subject => "10 ways to get traffic by writing blog posts about arcane Rails tips",
      :body => "Just kidding, it's totally impossible"
    }
    @request.env['RAW_POST_DATA'] = attributes.to_json
    post(:create)
    assert Article.last
  end
end

@request is a local variable defined inside of ActionController::TestCase, and used when you make a test request by calling post, get, etc.