As of right now Google is only showing that I have one site linked back to me and that is dmoz. Alta Vista and Alltheweb says I have 15 back links. Will Google ever acknowledge my other back links?

www.anchoraudioclub.com

Normally Google updates the backlinks and PR about once a month, give or take. Also they usually don't show backlinks which have a PR of <4, although I see more and more exceptions to this.....
But you don't have to worry, even if Google doesn't show your backlinks they will still be taken into account. The backlinks you see are just the ones Google is letting you see.