What Nathaniel wrote
*"This statement is over ${\mathbb Z}$. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either."* 
is strictly true in the sense that I do not know a manifold interpretation of the vanishing of the rational total surgery obstruction. However, the total surgery obstruction of an $R$-coefficient Poincare duality space can be defined 
for any ring $R$ with ${\mathbb Z} \subseteq R \subseteq {\mathbb Q}$. In the 1970's Quinn (and others) developed surgery obstruction theories for $R$-coefficient Poincar\'e duality spaces, but these theories have languished both for theoretical and practical reasons (e.g. a lack of examples). 

If anyone is interested in reading more about the total surgery obstruction, here are some documents which are available online:

[The total surgery obstruction (1978)][1] paper

[Algebraic L-theory and topological manifolds (1992)][2] book

[The total surgery obstruction (2010)][3] slides of MPIM lecture

  [1]: http://www.maths.ed.ac.uk/~aar/papers/total.pdf

  [2]: http://www.maths.ed.ac.uk/~aar/books/topman.pdf

  [3]: http://www.maths.ed.ac.uk/~aar/slides/mpim.pdf