The USPS official website says they're not supposed to take more than a value $20 and no cash or checks. However, I doubt this applies to real life.
It does apply to real life...explained below.
I mean I tip other professions, why not the mailman?
They are not professionals who are in a position to take tip-money (or be bribed), they are civil servants. I was a student worker at a state agency, and Starbucks sent one of our employees a $5 gift card. The HR lady and he agreed that it was not ethical for him to take it, and it was passed on to me because I was an hourly (contracted, not
civil servant) employee.
He actually does stand to lose his job for accepting a gift above $20, and most employees know better than to accept something more than this. If you want to show appreciation, some sort of home-made box of treats and a card or something similar would probably be the best way to go about it. I am sure he would appreciate something more expensive, but if it comes at the potential loss of his job and pension, he's probably going to have to respectfully decline it.
But that's just my four cents...