Upgrading to the newest stable version of org-mode (8.2.5h) made this easy-peachy. Now:
#+OPTIONS H:5
... actually works.
This works to get rid of the underlinings:
(setq org-ascii-underline (quote ((ascii) (latin1) (utf-8))))
In addition this contributes to get the clean look I was going for:
(setq org-ascii-headline-spacing (quote (1 . 1)))
(setq org-ascii-inner-margin 0)