In your last criterion, you are essentially asking for a "natural" problem that is nonrecursive, recursively enumerable, and is not complete for the recursively enumerable sets.  Post proved the existence of such problems in *[Recursively enumerable sets of positive integers and their decision problems][1]*, for many-one reductions.  Friedberg and Muchnik proved this also holds for Turing reductions, in separate papers *[Two recursively enumerable sets of incomparable degrees of unsolvability (solution of Post's problem, 1944)][2]* and *On the unsolvability of the problem of reducibility in the theory of algorithms*.  Whether these are "attractive" is probably determined by whether you like nonconstructive arguments.  For a clear and self-contained exposition of these results, see Kozen's book Theory of Computation.

So this is only a partial answer, and it would still be nice to exhibit a *real* problem with intermediate degree.

Edit: in the survey *[Degrees of Unsolvability][3]* (which appears to be a chapter of an unpublished Volume 9 of the [Handbook of the History of Logic][4]), Ambos-Spies and Fejer state "it is fair to say that every particular c.e. set of natural numbers that has arisen from nonlogical considerations so far is either computable or complete... Thus one could say that the great complexity in the structure of the c.e. degrees arises solely from studying unnatural problems."  This is quite a negative assessment!  On a more positive note, Feferman showed in *[Degrees of Unsolvability Associated with Classes of Formalized Theories][5]* that every c.e. degree arises as the degree of some recursively axiomatizable consistent theory of first-order predicate calculus.

  [1]: http://www.ams.org/journals/bull/1944-50-05/S0002-9904-1944-08111-1/
  [2]: http://www.jstor.org/stable/89817
  [3]: http://www.cs.umb.edu/~fejer/articles/History_of_Degrees.pdf
  [4]: http://www.johnwoods.ca/HHL/
  [5]: http://www.jstor.org/stable/2964178