I've been doing it for years, and never had a problem. You still receive your normal bills, or even view them on-line before you get them in the mail. You can take care of all problems beforehand.
Plus, some credit cards are not fully automatic. You still need to go on-line each month and select how much to pay. Others give you an automatic choice of full, minimum, or selected amount.
It's also useful for payments that you can't mail, anyways. I pay my rent through direct debit ($2 extra charge though.) That way, I don't have to print a check and drop it off at the office, assuming it's even open.