I figured out a good way to force tumbleweeds to spawn on demand.
Descend to the "Desert Entrance" bird statue. Head north to a sandy area. There are 3 landmarks of interest to us here. To the north of where you enter the sandy area is a small, round plateau with two bushes on top. A short distance south of the plateau is a dead tree. A short distance north of the plateau is a cactus. You will probably want to kill the monsters near these landmarks before continuing. Repeat the following steps until you have as many tumbleweeds as you want:
1) Climb the *south* side of the plateau. 2) Jump down on the north side and move towards the cactus. Keep an eye out for gusts of wind - that means that a tumbleweed should be coming up behind you on either side (catch it in your bug net.) 3) Climb the *north* side of the plateau. 4) Jump down on the south side and move all the way to the dead tree.
This takes advantage of very fiddly game mechanics which I don't fully understand. It may take a cycle or two to get started, but if you're doing everything exactly as laid out above, you will hopefully get a tumbleweed almost every cycle. If you're not getting consistent results, make sure you're following the directions exactly (even climbing the wrong side of the plateau can throw everything off.) Let me know how it works out for you - my instructions may need clarification, or it may work differently on certain versions of the game.
Eh, I always got my tumbleweed at the North Desert area (or wherever the Silent Realm/Mining Facility Entrance is) and just stayed close to the stairs and every 30 seconds or so, the gust brought a tumbleweed. (With the Treasure Medal equipped)
C-A --- Cloud and Sephiroth - battling each other since '97 for your amusement Give it a rest already!
Ahh... I remember chasing tumbleweeds across the desert while getting attacked by those electric crab things... That never ended well... --- https://i.chzbgr.com/completestore/12/8/11/wXLYp87DiE2C6xnGXLS7BA2.jpg