In the case of the end-of-year mailing of payments that didn't arrive in your hand until the following year, despite the date on the check, you didn't constructively receive the money until the following year. So it's taxable income, in most cases, in the year you got the check.
Keep in mind, the person who sent you the 1099 is probably going to provide the 1099 to the IRS. But if you have good records of the date you received the money, you will have no problems.
