Instead of shoving piles of cloth placemats and napkins in a drawer, organize them in sets by using extra-large zipper-sealed bags. Place the set in the bag and use a marker to write how many of each are in the bag. The bags can then be stacked in a large drawer to keep everything neat and organized.
If you are traveling and need to bring something critical, consider shipping it instead. Many items cannot be carried on due to limitations or size. Instead of checking your valuable item in, ship it ahead of time to someone you trust at your destination.