If it's a persistent issue, the best move is probably to write on a sticky note "return to sender -- please make note to deliver only [names] to [address]." If it doesn't happen often, you could probably just write "return to sender" on it. Also, you can refuse any piece of mail, so if a certified letter or something for a previous resident comes to you, you can tell the carrier that they've moved (please don't abuse this if it's actually for you, though).

If your mailbox has a flag on it, you could just leave the flag up with the letters for the previous tenant and the carrier will (rather, ought to) take them away. You could also tape a note inside your box saying "mail for [cassieee] only", though notes like that tend to fade and are probably the worst solution.