Yes, there are a few non-secret fraternities, but none are as big as Delta Upsilon, which is the largest and oldest non-secret fraternity.
We were founded at Williams College in 1834 -- we are the sixth oldest fraternity in North America. From The Cornerstone (our manual), "DU had no need for mystic principles shrouded in secret ritual. Our aims were open, honest and direct."
Personally, I feel that being in a non-secret fraternity has its advantages. Not only do we get to experience and live a life like every other fraternity, but we also get a chance to show everyone else what we are about. Our initiation is open to public scrutiny. We invite parents, friends, and university administration, so not only do we have great brotherhood, but others like faculty know that we do not have anything to hide and that allows us to sometimes have a higher level of respect with them.
I'm not saying that a non-secret fraternity is better than any other, don't get me wrong. It is just different, that's all. But I see no disadvantages, but sure, I'm obviously biased
You can see pictures of our initiation and other stuff on our website,
www.mudeltau.org. It takes place on the front lawn of our Chapter house, which is unique to our campus.
Brian