Yes, it's possible to be morally upright without being religious. Partly, it's because non-religious people have the religious heritage there in society to draw on, even though they don't accept its basis. Partly because, as St. Paul says, "the law of God is written in the heart"-- What's known as Natural Law.
I just did a Google search for that phrase and turned up a couple of good articles and books on the subject by writers I know, particularly J. Budzewzski and Phillip Johnson:
http://www.google.com/search?sourceid=navclient&ie=UTF-8&rls=GGLG,GGLG:2005-46,GGLG:en&q=law+of+God+is+written+in+the+heart
But without the support of religious faith it's a lot harder.
It is harder, unless well, you are raised right. Then it is easy. It is hard wired. It is as natural as breathing. It is who one is. Is is all about good habits, in short.
Nice post.
Budziszewski and Johnson, both very good choices. I'd also toss in Charles Rice's book 50 Questions on the Natural Law: What It Is and Why We Need It along with pretty much everything on this fellow's Amazon list on the Moral Law .