A lot of folks don't think about the items in the safe deposit box or think about banking needs at all during the move.
Small valuables such as jewelery, insurance policies, stamp collections, etc, can be carried with you or send them by insured or registered mail.
Same goes for items that may be in your safe-deposit box. Most folks don't think about this until they have need of them after their already in a new town. Save yourself some hassle and deal with that now.
Arrange to move your deposits have new bank accounts setup before you move. Ask your current bank about the best way to handle transferring your funds to the new location; maybe they have a location to serve you there!
Another seldom thought of valuable is all the data on the hard drive of your PC. Hard drives are fragile and do not need to be slammed or banged around. Backup your data to an on-line provider or pull the hard drive out and take it with you when you go. Don't risk throwing it in the moving van and hoping it will make it.