While bows could be simply carried around the shoulder, a practical method of carrying them had evolved on the Steppe consisting of a large holster with a quiver on the side. This bowcase is referred to in Greek sources as a gorytos (pl. gorytoi, Lat. gorytus). It lasted for hundreds of years before being replaced with the separate bowcase and quiver of Turkish and Mongol horse archers.
The Achaemenid gorytos consisted of a rectangular, presumably leather pocket worn diagonally, bottom-forward, at the left hip. Its forward corner was rounded off. It was probably one piece of leather, folded and stitched along the upper edge with what looks in some illustrations to be a Holbein stitch, though in others a separate creased strip runs along the top edge which I am not sure how to interpret. The strung bow rested in the case staff-down, and the curved corner follows the curve of the bow's tip. This left the bottom corner of the gorytos empty. A good overview may be seen here (second from the bottom on the left).
Lightly moisten a line down the middle of the case and cover pieces, on the flesh side. Fold them, smooth side out, and place weight on top so they'll dry in the right shape.
Make stitch holes with an awl, leather punch or small drill bit as the thickness and hardness of your leather requires. Using heavy sinew or string (about four times the length of the seam), first stitch on the arrow pocket and then close the bowcase and cover seams.
My own solution to this problem was to rivet down a piece of leather through which a leather lace is threaded.
The holes for the lace are punched into the intermediary leather piece first. Then the bottom three rivets - solid ones with washers - attach the intermediary to the unstitched case below the line of stitch holes, being hammered down on the inside of the case, and the lace is threaded through the intermediary. After the case is sewn, the upper two rivets are installed; these are longer rivets that go all the way through the case above the stitching.
It doesn't look good, but with larger, domed rivets covering most of the intermediary piece, it would probably look something like the originals.
It won't fit there
If you bowcase or cover are too tight for the bow, you can expand them by dampening and filling them with various objects, such as wooden boards and dowels, wherever they need to loosen so the bow can be removed easily when you go to shoot. Don't use loose matter (the way you would expand a leather water bottle with grain or sand); you don't want the case to bloat like a balloon, but simply to expand in the necessary places while maintaining its shape. I recommend doing this especially for the top end of the cover, where the curved tip of the bow fits rather snugly.
It's also essential for the arrow pocket to hold the arrows loosely (including large blunt safety arrows) so you can draw them fast. Put a board (a large book works okay if you're careful not to make the arrow pocket too wet) into the bowcase to hold it taught, then fill the pocket with dowels, wiffle bats, just about anything to give it a full shape, filling in as evenly as possible to prevent lumps. Let the leather dry completely before removing the objects.
Next up: The Persian army.