Complexity class separations without hierarchy theorems
I'd love to be shown wrong, but I don't think there are currently any uniform lower bounds that aren't ultimately based on one of the hierarchy theorems. Our current understanding of how to take advantage of uniformity is really quite limited in that sense.

On the other hand, there are many uniform lower bounds that don't follow directly from hierarchy theorems, but use a hierarchy theorem in combination with other clever tricks, techniques, and results, for example: 


$\mathsf{CSL} \not\subseteq \mathsf{DTIME}(n)$ [Hopcroft-Paul-Valiant]. They prove that $\mathsf{DTIME}(n) \subseteq \mathsf{DSPACE}(n/\log n)$ (the non-diagonalization part of their proof), and then use the fact that $\mathsf{CSL} = \mathsf{NSPACE}(n)$ in combination with the space hierarchy. Their result + the space hierarchy also implies $\mathsf{DSPACE}(n) \nsubseteq \mathsf{DTIME}(n)$.
Time-space trade-offs for Satisfiability (see, e.g. the introductions of Buss-Williams and references therein)
$\mathsf{DTIME}(n) \neq \mathsf{NTIME}(n)$ [Paul-Pippinger-Szemeredi-Trotter]. Uses a nontrivial simulation of any deterministic super-linear-time machine by a faster four-alternation machine, in combination with the deterministic time hierarchy.
Uniform lower bounds on the permanant [Allender, Allender-Gore, Koiran-Perifel]
$\mathsf{NEXP} \not\subseteq \mathsf{ACC}^0$ [Williams] (although technically this is a nonuniform lower bound, it uses a bunch of clever ideas in combination with the nondeterministic time hierarchy)