Attack the enemies you need to defeat in order to collect materials - your quest log will tell you which enemies you need. Bring a friend or two if you can in order to bring in four enemies at a time (will double the speed of gathering). Defeat the enemies in the duel ring, and pray that you are lucky so you obtain the quest drops.
Rule of thumb: The more items you need to gather, the less likely you are to not gather anything. If the amount of items needed surpasses 10, you are pretty much guarenteed to get one item per enemy. If the number is 3 or less, settle in for a long time battling the same enemies over and over...
