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)?



Undecidable theories easier than $Q$
When I was looking around trying to find some inspiration to answer your question, I found the following result of Feferman from 1957:



> 
> For any set $X$ of natural numbers there is a theory $T(X)$ such that:
> 
> 
> * The set $X$ and the set of Gödel numbers of consequences of $T(X)$ have the same degree of unsolvability.
> * If $X$ is r.e. then $T(X)$ is effectively axiomatizable.
> 
> 
> 


Because there are nonzero r.e. Turing degrees strictly weaker than $K$, I think this may answer the question. 


The result is in the paper "Degrees of Unsolvability Associated with Classes of Formalized Theories", Solomon Feferman, *The Journal of Symbolic Logic*, Vol. 22, No. 2 (Jun., 1957), pp. 161-175. <http://www.jstor.org/stable/2964178>


