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.

  [1]: http://www.ams.org/journals/bull/1944-50-05/S0002-9904-1944-08111-1/
  [2]: http://www.jstor.org/stable/89817