Most proofs of undecidability for various theories (pure logic with binary relation, group theory, etc.) show that the natural numbers and Robinson's $Q$, in one form or another, can be encoded appropriately. Hence the decision problem for these theories is as hard as $K$, the halting set.
Are there are recursively axiomatized theories which are undecidable, but yet easier than $K$ (i.e. $K$ is not Turing reducible to deciding to the theory)?