What happens if I cast disenchant in response to the triggered ability of animate dead?
If you destroy Animate Dead before the creature card it enchants returns to the battlefield, the creature card will remain in the graveyard and never enter the battlefield.

Read the text of Animate Dead carefully:


  When Animate Dead enters the battlefield, if it's on the battlefield, it loses "enchant creature card in a graveyard" and gains "enchant creature put onto the battlefield with Animate Dead."


If Animate Dead enters the battlefield, it triggers its ETB ability. If someone responds with a Disenchant to destroy it, then by the time the ETB resolves, the aura will no longer be on the battlefield and the ability does nothing on resolving.

The Oblivion Ring trick you mentioned works only because its ETB ability does not have that condition.