By Don Bluth:
The Secret of NIMH
The Land Before Time
All Dogs Go To Heaven
Anastasia (yeah, its like Disney, but its still pretty damn good)
Titan A.E.
By Dreamworks:
Prince of Egypt (really, I think its just alright, but its still quite good)
Spirit: Stallion of the Cimarron
Sinbad: Legend of the Seven Seas (weak near the end, but very fun)
By Warner Bros:
Iron Giant
Cats Don't Dance (flawed but with terrific animation)
Corpse Bride (again, a bit weak near the end, but has incredibly smooth stop-motion)
A Scanner Darkly
The entire works of Aardman studios.(i.e. Wallace and Gromit)
By Henry Selick:
The Nightmare Before Christmas (only the most minor of Disney influence)
James and the Giant Peach (I don't really like this movie, but the animation is great)
Coraline
Final Fantasy: Spirits Within is just alright, but its CG is still wonderful.
Final Fantasy: Advent Children COMPLETE (seriously, the blu-ray cut is much better, as it makes the movie make more SENSE)
Blue Skies:
Ice Age (an okay movie, but it has a unique visual style and setting, with believable casting)
Robots (very weak near the end, and the story isn't that great, but it is a true feast for the eyes)
I recommend the works of Satoshi Kon. (Perfect Blue, Millenium Actress, Paprika, etc.)
As well as the works of Studio Ghibli, though I'm not as big a fan of them as many others are. However, Castle in the Sky is my favorite of their releases.
Most of the Bruce Timm-produced DC animated movies are quite entertaining, as well.