Suppose $M$ is a $3$-manifold. Then you can split it over spheres. This is the "prime decomposition" and is unique. You can then split the components of this decomposition along tori. If you leave the components which are Seifert manifolds alone then this is called the "JSJ-decomposition" and again is unique.
My question is this:
Can you ever split over tori if $M$ is not a hyperbolic manifold?
I am sure I read this somewhere, but I have been through, over and under all of the things I have been reading about them and cannot find anything which mentions this. So...help? A reference would be great, and so would a short proof. Both would be best of all!
(Oh, and if you know what tag this should be under that would be great! I am figuring general topology isn't quite right, but then neither is algebraic topology nor differential topology...)