I couldn't make out exactly what your beads depict, but taking your word that the lines represent a dog, perhaps it's the howling coyote or wolf - Google images search:

https://www.google.com/search?q=coyote+turquoise+bead&es_sm=93&source=lnms&tbm=isch&sa=X&ei=-3wEVYC6CouuogSVpIHgDg&ved=0CAgQ_AUoAg&biw=1280&bih=685#tbm=isch&q=carved+coyote+turquoise+