I set myself the goal to spam each specific dungeon (in ascending order of level) until I have all the green loot from it before moving on the next one, changing class to be able to Need the pieces I was missing.

For example,

Sastasha/Copperbell/Tam-tara (Foestriker, Plundered, Acolyte) > Halatali (Doctore's weapons/armor) > Toto-rak (Warden/Brigand/Torturer/Ascetic's) > etc.

By the time I got to Aurum Vale (lv47 loot), all my combat classes were lv50 without whatsoever need of FATEs and roulettes, though I do not recommend this goal-setting kind of approach if you don't have a collector's OCD like me who actually makes me enjoy the repeating of the same dungeon several times a day.

DoHs, just use the HQ bonus on Supply Delivery and crafting leves (www.fxivguild.com helps). DoLs were the ones that look longer for me, but the Supply Delivery HQ exp bonus and the gil income during the grinding made up for the long time it took overall.