Recycle's ability to make a deck last longer in a battle makes it essential for every deck. If you have a deck without at least one Recycle creature, either you're ballsy, or ignorant.
The one weakness that Recycle has is that it never brings cards back to your hand, so it may take some time before the recycled creature can be played. This allows for the enemy to get field control, and possibly take you out.
It is also possible to interfere or disturb Recycle with the Torment skill found on the Paragon and Geomancer. Another way to fight Recycle-heavy decks is by using a Hero with Ghoul Feast, Reanimate, or Corpse Explosion, which can be found on Overlord, Armageddon, and Balrog, respectively.
Not really a "weakness" but still something to consider about recycle is that the ability will often make your enemy attempt to dispose of the creature (if they're able to manually control the battle). Because of this, the ability is often melded onto a creature with either stoneskin or immunity (most commonly Horned Beast).
Also, one thing to consider when adding Recycle creatures to your deck is not to add too many of them. Typically, most people go with either one or two Recycle creatures, as having too many may result in your deck being too weak.
Note: Although the new Skill Enhancement feature allows you to upgrade your creatures' skills, Recycle is not available as an upgradable skill.