Neoverse

What you need to do is NOT pick up and evolved cards in any shape or form during your run through to 3-5 (or farther if using Unknown Stone). Do not get any as battle rewards, buy any in the store, or even pick any missions that give a reward of evolving a card or evolved card. Technically you can take these missions and complete them as long as you don't accept the reward for completing the mission but why risk accidently accepting it.
Evolved cards add a slight power up to that card in some way and there are a set of wings on the top border around the gold diamond that displays the mana cast cost. Again, pick none of these up.

I just got this using the Snorkeling Goggles and it didn't void the achievement.

I used Claire (Hybrid Deck) and the three pieces of gear I equipped were:
• Snorkeling Goggles - Randomly evolve a card at the beginning of every turn.
• Backpack - +1 Mana at the beginning of every turn.
• Little Undead - You can swap the cards in your card rewards to see another 3 once each card reward if you didn't like the first 3.

Every time the Goggles evolve a card it stays that way the rest of the combat so the longer the battle the more evolved cards you'll have. It's random of course but having all of those upgraded cards as the turns tick by make this achievement really easy to get.

I did have one evolved card but I hung on to a "remove a card" mission reward and I removed it at the end of level 3-4.

All the gear pieces have to be bought with Trophies (this game's currency) and they can be really grindy to obtain since you can only get one from beating a boss I think. However I happened to buy a DLC costume pack for two bucks and unexpectedly it also gave me 100 Trophies. (I bought the Super Heroes pack but I assume they all give 100 Trophies though.) If grinding out Trophies is too slow try getting a DLC pack (there's four) and you'll have enough to buy all the best gear including the Unknown Stone if you plan on doing the Transcended Universe levels.