Yes! But you have to watch out for Amex, for example. If you get a new card, no new merchants can charge to the card, but they will allow charges from merchants who have previously charged to the card.Report the card lost to the CC provider and get a new card with a new number.
Amex even allows charges for so many months on an expired card.
Also with DIRECTV they have 2-3 different accounts they use to process charges when your on auto pay. Try calling and placing a block on DIRECTV, they will just charge under a different merchant account with the same name the following month. I don't know if Dish has the same practice, but it wouldn't surprise me as its usually monkey see monkey do with these two providers.