Camera screens can get scratched, no doubt about it, and I'm debating whether to purchase a protector, although the cheaper ones seem to have issues with coming unstuck, and the more expensive ones are...well, expensive.
As a temporary solution I've put a lens cleaning cloth (a regular silk type available from opticians) in the case with my camera. I folded it in two and laid it flat along the 'long end', the part where the screen sits.
I figured that grease and any other bits we don't want will only be transferred to the inside of the case after prolonged use, and you can't really clean it. However I can wash the cloth every couple of weeks and it is handy to polish the screen with.
Not known for my genius ideas, but this might help someone!