First of all, the things that you actually integrate  are  densities, which  are the differential geometric counterparts  of  measures.   No orientation is needed.   On a smooth manifold $M$ with separable topology there is an intrinsic concept of negligible  set.  A density is then a signed measure $\mu$ such  that $|\mu|(B)=0$ for any negligible Borel subset.

 A degree $n$ form on an $n$-dimensional manifold is almost  a density, but not quite. We need an orientation to  associate to the top degree form a  density.  This  is what you ultimately integrate when you integrate  a  form.  For more details see  Section 3.4.1 of [these notes][1].


  [1]: http://www.nd.edu/~lnicolae/Lectures.pdf