No, it is only ever harder.
Just as the decision problem for intuitionistic propositional logic is harder than of classical propositional logic, so to is linear propositional logic harder still. With either exponentials (which don't lack contraction) or various flavours of noncommutative connective, the logic becomes undecidable and even the weakling classical MALL is PSPACE complete. By contrast, the decision problem for classical propositional logic is co-NP complete, and for intuitionistic propositional logic, PSPACE complete. (Offhand, I don't know the complexity of intuitionistic MALL.)
I recommend Pat Lincoln's exposition in section 6 of his Linear logic, SIGACT News 1992. We have learned a bit more since then, that is, we have results for a large family of linear logics, but the basic picture is there.
In a certain way, this is what makes proof search for linear logic interesting, since hardness of the decision problem makes space for more interesting notions of computation, and linear logic is hard in so many different ways. Andrej pointed to Dale Miller's An Overview of Linear Logic Programming; this is a good place to look since Miller has done more to develop the idea of proof search as computation as anyone else.