Removing grease and fingerprints from the screen can be a bit of a challenge. Cleaning it with you shirt cuff doesn't really work. Jumpers and sweatshirts tend to be pretty useless as well. You can use tissues but they're not much more effective than wiping an inverted iPad over your jeans, which doesn't do much more than smear the grease. Oakley sunglasses bags work reasonably well but take a fair amount of effort. I haven't tried carpets but I don't think I'd advise their use. Soft velvet curtains might work and you might find one in your local National Trust property, but that's not a very practical solution (and the NT room stewards can be a bit scary if they think you're damaging their charges).
So what are your remaining options? I've found that the best tool for cleaning the screen of my iPad is the cleaning cloth supplied with our LG LCD TV, which works really well. No idea what it is made from, so this advice is not going to be uniformly helpful, but if you've got an LG LCD TV kicking around you should find that the green cloth they include in the box is really good and fits easily into many iPad cases for handy carrying.