Backpack Hero

Cursed items are typically bad items that will follow you around until you finish the game or find a way to remove them. Once you have a cursed item, you either have to equip it, or else it will appear in every battle reward screen.

Finding cursed items is not difficult, but it is probably best to wait until later in your Story Mode campaign to give this a try. Mostly because there are more Cursed chests in lower floors.

First, we will need 4 cursed items.
As mentioned, cursed chests (the black and red ones) will always have one cursed item. These are never locked either.
Furthermore, many '?' rooms may offer ways of getting cursed items.
The dice rolling one is a sure bet, as the penalty for over-rolling is a cursed item. You can just keep rolling until you get it.
The card memorization one is good too, as it sometimes has multiple cursed items.

Anyways, when you get four, add them all to your backpack and the achievement will unlock. You can immediately remove them, so don't worry about having to play with them.

If you want to salvage your run after the achievement, there are a couple of ways to get rid of curses. After a boss fight, the healer will offer to remove curses for a fee.

In addition, I found a method (possibly a glitch) to remove all cursed items. It requires you to find the dice item. Once found, keep it in your backpack.
The next time you finish a battle, the cursed items will appear (as normal).
At this point, use the dice (press X on the item, and then select 'alternate use'). This will re-roll your battle rewards. However, in this case, it will clear out all of the cursed ones. It even removes the curse counter from the top of the screen.