The problem is that there is no way to figure out signs - It would be like trying to integrate a function from $\mathbb{R}$ to $\mathbb{R}$ without knowing whether you were moving forward or backward. 

What you CAN actually integrate are pseudo-differential forms.  The whole point of choosing an orientation is to turn a differential form into a psuedo-differential form.  For those, I recommend the wonderful short story by John Baez found here:

https://groups.google.com/group/sci.physics.research/msg/3c6a1a7237b66c8c?dmode=source&pli=1