In the northern parts of SA there are some great signs that say 'welcome to the Northern Territory' those are my favourite..
Not offroad, but Hahndorf is a beautiful area east of Adelaide, little German town. I'm certain there is some offroad around there, but it's all likely private property.
Have heard of a place called Humbug scrub too, but never been there myself.
http://youtu.be/NCqpDX3T30Q