[Rowebot]

Most dyes are created from items that can be farmed or produced with bone meal.  The 4 double height flowers can be duplicated by right clicking with bone meal -- each right click drops one duplicate flower to pick up.  Lapis for blue is the hardest to get a steady supply of; unless you mine for it regularly.

So let's go through them from http://minecraft.gamepedia.com/Dye and look at the most sustainable ways to make them:

Red -- get a beetroot farm or use bone meal on a rose bush flower

Orange -- use bone meal on a sun flower to get yellow and combine with red dye

Yellow -- use bone meal on a sun flower

Green -- put cactus in a furnace

Blue -- lapis lazuli is sustainable by trading with villagers; else mine it with a Fortune III pickaxe for maximum drops

Light Blue -- craft blue + bone meal; else bone meal Blue Orchid areas in swamps

Magenta -- bone meal a Lilac flower

Pink -- bone meal a Peony flower

White -- bone meal

Light Gray -- 2 bone meal and 1 ink sac

Gray -- 1 bone meal and 1 ink sac

Black -- ink sac

Brown -- Cacao Beans

Cyan -- green + blue

Purple -- red + blue

Lime Green -- green + white

So you need to build or get access to the following farms:

  Skeleton spawner for bone drops -- most important farm

  Squid farm for ink sac drops -- many guardian farms produce these as a by product

  Cactus farm for cactus

  Cocoa bean farm for Cocoa

  Villager trading center for Lapis or just mine for it on a regular basis

If you can find a skeleton spawner under an ocean, then set it up for afk farming for drops, build a squid farm above the spawner, an automated cactus farm floating above the squid farm, and a cocoa bean farm under the spawner.  Optionally plant beetroot under the cocoa bean level.  Buy VIP1 for unlimited afk and park your butt at the spawner.  Get one each of the double height flowers and you are set for all but Blue and Light Blue.  You have the option to use bonemeal in the right place in a swamp for blue orchids to get light blue.

End result -- you get all dyes for free except Blue for parking your butt then doing a bit of manual work: harvest, replant, bone meal flowers, use a furnace and use a crafting table.  Another example of Minecraft mimicking real life -- before synthetic dyes, blue and purple were the hardest dyes to get.

[BlueTriforce]

Thank you for this post! I'll reference it frequently~

[The_4thdoctor]

Lucky me, i have a skelly spawner, beetroots, lots of lapiz and flowers are easy to find with /random