When a ruminant chews cud, it comes up from the first stomach. Items that the body decides cannot be digested move more quickly through the digestive system, and yes can come out nearly intact. Spend any time on a working farm - not a bug commercial farm where feed is always controlled, but a small family farm where pastures are multi-use and animals eat many things they probably shouldn't and you will see this.
It's perfectly plausible that linen based money will pass through a cow relatively intact.
OK - and doing a partial U turn - I'm convinced on what Thewitt says. But wouldn't that be with occasional items?
And having thought about it properly, why would a cow eat a lot of money in the first place? So one bill, a few bills, mixed in with feed, coated with molasses - that might go down and be in with the general tummy slurry. But the quantities of money the OP is talking about - would the cow eat that much?
If the cow has eaten lots of stuff it can't digest, does the send it on through/send it up for chewing system cope with mostly send it through?
Though having said that I just remembered this story
http://news.bbc.co.uk/1/hi/england/west_midlands/3614022.stm
Canvas can be linen and cotton. The aircraft bits clearly had something tasty on them. However, just because the cows bit off all the canvas, there is a chance they didn't swallow just wandered off and dropped it. No information on whether the canvas showed up in the poop.
And then I watched the video with the model airplane that also came up in the search
http://fpvcentral.net/2013/11/cow-eats-fpv-plane/
You can see a cow chewing on a bit with wires hanging out of it, but otherwise largely licking. I think they were youngsters - not sure. Do remember a neighbour saying that a calf will eat anything until it knows better and he has to make sure there are no bits of baler twine in the field for example as that can cause problems for the calf it it eats it - goes in part way and gets stuck.
OP - you might want to do searches on laxatives on cows and the like. Am wondering whether you'd give laxative of the prune variety or paraffin oil as you give to cats to help them pass a hairball.