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) paper
Algebraic L-theory and topological manifolds (1992) book
The total surgery obstruction (2010) slides of MPIM lecture