You have to use three different mouths which can be placed anywhere in the body. Note that this is a Creature stage achievement, so using the Creature editor outside of the game or in the Space stage will not count.
If you try to make your creation look like the achievement's picture with three heads (for example, by placing a mouth at the end of a matched pair of arms), that won't work. You will have three mouths, but only two different types of mouths (unless you use the Asymmetrical feature).
You will technically need three, four, five, or six heads, depending on placement. Three if all are along the center; four if one is not along the center of the creature; five if two are not along the center of the creature; six if none are along the center of the creature.
With a mod or the patch 1.05, you can use asymmetrical placement and create three different heads like in the achievement's picture, it will work for this achievement.