We explore ways to embed a ternary tree in an integer coordinate grid such that the width of the drawing is minimized. We provide upper and lower bounds on the width requirement of planar, straight-line, upward, order-preserving drawings of ternary trees in an octagonal grid. We present a linear-time algorithm for constructing such octagonal grid drawings of any n-node ternary tree with O(n0.68) width. (This bound can be improved to O(n0.631) width in the so-called HVA-model.) For ideal octagonal grid drawings of complete n-node ternary trees, we provide an Ω(n0.411) width lower bound