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

EDIT:

The story is too good to miss, so I am adding the text to my answer.  I know a lot of people don't like to click on links.

________

It was a snowy Christmas Eve.  As the wind howled outside the
castle tower, the Wizard was playing his flute by the fireside,
meditating.  All the final exams were graded, the apprentices
had all gone home, and he had spent an intense week working on 
Lie n-groupoids and a thought experiment which showed that quantum
gravity effects made it impossible to measure distances with
an uncertatinly less than Planck length.   Now it was time to
relax!   The fire crackled, the wind whistled down the chimney,
the flute's melody swirled like the smoke... it was all very 
peaceful until he heard someone pounding on his door.  

"Whoever it is, go away!" shouted the Wiz.

A muffled voice came from behind the door.  "It's me!  Eric!"

"All the more reason!" growled the Wiz under his breath,
putting down his flute and walking to the door after giving
the fire a poke.

"I come seeking knowledge!" 

"Oh yeah?" said the Wiz, heaving the oak door open on a crack
and sticking his nose out.  "What exactly do you want to know?"  

"All I'm asking for is reasons!" cried Eric.

"Reasons for what?!" 

"You keep saying I am just being ignorant, but I'm holding out 
until I see some honest to goodness reasons why pseudoforms are 
necessary."

The Wizard began to fume.  "Not that again!" he said.  He tried
to slam the door shut, but Eric had already slid his foot in the
crack.

"Ow!  You're not getting rid of me that easily!"

"Hmm," said the Wiz and pondered a while.  He couldn't help
admiring Eric's persistence.  He'd once had an apprentice named
Oz, whose education was woefully inadequate in every way, but
who managed to make real progress nonetheless, thanks to his almost
insane persistence.  Unfortunately, Oz had gotten swallowed by 
another universe as part of a disastrous experiment, and so far 
all the Wizard's attempts to recover him had failed.  Out of pity 
and fond memories, the Wiz decided to give Eric one more chance.  
It *was* Christmas Eve, after all....

Peering through the crack, he started with a sneaky question: "What 
sort of thing can we integrate over any smooth n-dimensional manifold?"

Eric cried: "An n-form!  This is why differential forms are
so great!  Everything we actually measure, physically, is the
integral of some quantity over some region, or curve, or...."

"Wait," interrupted the Wiz.  "Before you launch into your
personal philosophy again, let's get the facts straight.  It's NOT
TRUE that you can integrate an n-form over a smooth n-dimensional
manifold.  What you can integrate is a PSEUDO n-form."

Eric's mouth snapped shut in mid-rant, and then fell open again
in shock.  "Hmm... you've got my attention now!" he said.

"Good," growled the Wiz.  "So, listen up: You can
only integrate an n-form over a smooth n-dimensional manifold
if it is equipped with an ORIENTATION.  You may be so used
to this that you've come to accept the orientation as an inevitable
prerequisite for integration.  But it's not true!  Integration
of pseudo n-forms works perfectly fine on any smooth manifold,
even an unoriented or unorientable one.   It's only if you make
the mistake of trying to integrate an N-FORM" - he practically
spat the term out in disgust - "that you'll need an orientation.
And all the orientation does is let you convert your n-form to
a pseudo n-form!  Correcting one bad move with another...."  He 
trailed off, grimacing at the folly of the world.

"Hmm... this sounds very interesting," said Eric.

"Good.  More to the point, it's the truth," said the Wiz.  

"Why didn't any of the 10+ books I have on differential geometry
and forms mention this?!"

"What do you expect if you read so few books?!" fired back the
Wiz.  "You get what you pay for.  But surely even those miserable
tomes said you could only integrate an n-form on an n-manifold
that was *oriented*.   No?   And if so, surely it was your job
to wonder whether it was possible or not to do integrals on an
unoriented manifold!  You should have pondered a Moebius strip, 
and asked yourself: `What's to prevent me from doing integrals
on this thing?'  You'd chop it up into coordinate charts, do
the integral on each piece with the help of a partition of unity,
and then add up the results....  And after a little thought, you 
would have discovered the answer: it all works perfectly fine, 
*if* you stick in an extra minus sign to describe how your n-form
transforms under a change of coordinates whose Jacobian has negative 
determinant!  In other words, you should use the absolute value
of the Jacobian, just like the change of variables formula in
multivariable calculus tells you to!  But when you do this, you've
reinvented pseudo n-forms.

And then, after asking around a bit, you'd have discovered that
everyone... everyone who counts, that is... has already realized 
this!"   

Eric pondered a moment.  "But... but if it's *that* simple, why
don't my textbooks talk about it?"

Grinning evilly, the Wiz replied: "They certainly drop the
necessary clues.  As for why they don't *emphasize* this stuff, 
well, this is just one of those tricks we Wizards use to distinguish 
the people who think for themselves from those who fall for any 
plausible line of claptrap.  In fact, every time Halloween falls 
on a full moon - like this year - we get together and agree on 
what facts like this we will keep secret, precisely to see who
rediscovers it for themselves.  This year we...."  At this point 
the absent-minded Wiz caught himself.  "Whoops!  This is too
secret for you!  Anyway, you just failed one of these tests."
 
Mortally offended, but (admirably) more interested in the math 
than his own dignity, Eric replied "You might shoot me, but you 
are actually starting to make me think that perhaps it is
pseudoforms that are more fundamental than regular forms!"

"Great!" said the Wiz.  "Actually, this will *relieve* me
from the need to shoot you, or, more likely, turn you into a toad.
Anyway, I hope you see now that people who think they're really 
integrating n-forms are just slightly less naive than the people 
who think they're really integrating functions.   You can only 
integrate these things if your n-manifold is equipped with extra 
structure... and this extra structure is only being used to convert 
these things to pseudo n-forms!"

"Well," admitted Eric ruefully, "I certainly consider myself to 
be naive.  Honestly, I never knew this.  I would really like to 
understand this better.  Got any references?"

"References?!" cried the Wiz.  "What do you think I am, a walking
journal article?  I give you all the clues you need to figure everything
out for yourself, and you want *references*??  Here - HERE's your
much-beloved REFERENCES!" he said, flinging a fireball at Eric.  

The fireball was large and blue-white, hissing furiously.
Luckily for Eric, his reactions were quick and he ducked in 
time to miss being fried.  It hit the wall behind him and 
exploded in a bang.  Simultaneously, the Wiz pushed Eric's
foot out from the door and slammed the door shut.  

Then he opened a small flap and poked his face out for a
moment.  "Oh, and by the way: Merry Christmas!"  He tossed 
out a small candy-cane and snapped the flap shut.

Eric was left with his questions.  Sucking mournfully on the 
candy, he stumbled down the castle steps and out into the 
chilly night.

"Oh, oh!" he wailed.  "Can't ANYONE give me some references??"