This paper presents efficient structure-preserving signature schemes based on assumptions as simple as Decision-Linear. We first give two general frameworks for constructing fully secure signature schemes from weaker building blocks such as two-tier signatures and random-message secure signatures. They can be seen as refinements of the Even-Goldreich-Micali framework, and preserve many desirable properties of the underlying schemes such as constant signature size and structure preservation. We then instantiate them based on simple (i.e., not q-type) assumptions over symmetric and asymmetric bilinear groups. The resulting schemes are structure-preserving and yield constant-size signatures consisting of 11 to 17 group elements, which compares favorably to existing schemes relying on q-type assumptions for their security.