I'd never heard of the bread trick--thanks for that, gregory and tarotbear!
Depending upon the gunk, the paper, and the finish, my go-to's for cleaning soiled paper:
1) white "mars" eraser
2) spot clean using isopropyl alcohol and a cotton ball or cotton swab (look for 90% or greater)
3) spot clean w/clean cloth dampened in clear warm water
4) spot clean w/clean cloth dampened in sudsy water (Dawn dishwashing detergent recommended)
I've also had success with the white vinegar process closrapexa recommends.
ADDED: If you think may be an oily stain, try placing the card on a white paper towel, sprinkle white flour or cornstarch over the stain, put another paper towel over it, and put a couple of heavy books on it. Leave it overnight, check on it, and replace the flour and repeat if necessary...unless it doesn't look like it's helped at all.
good luck with your project!