I worked for many years as an attorney for the Postal Service, and can assure you that the Standards of Ethical Conduct for Employees of the Executive Branch (Title 5, Code of Federal Regulations, Section 2635) do NOT permit federal employees - including postal employees at all levels - to accept CASH in any amount, or any other "gift" with a market value of more than $20. However, my office waged a mostly-losing campaign for years at getting the media (and our own employees) to recognize these gift prohibitions. The reason for the prohibition against gift-giving of anything with a value of over $20 is that government employees are already paid to do their jobs -- in the case of letter carriers, that is to provide excellent service in delivering the mail.
If you are pleased with your carrier's service, feel free to leave a holiday tin of cookies, or a gift card with a value of $20 or less, or an ornament, or anything else (except for alcohol - never permitted) valued at $20 or less. Better yet, send a letter to your postmaster (the address is Postmaster, your city's name, your 5-digit ZIP code with the 4-digit add-on of 9998), and let the postmaster know how much you appreciate your carrier's service. Yes, you need a stamp, even to send a letter to the postmaster :)
By the way, carriers who are caught accepting monetary gifts from customers can be disciplined for violating the Standards of Conduct , and/or charged with illegally supplementing their federal salary. So, you are NOT doing your carrier any favors by leaving money (cash, check, gift cheque) for him or her. Any carrier (or any OTHER postal employee) who solicits customers or vendors for money, at the holidays or at ANY time, especially if the solicitation implies that you will receive better service or special treatment if you provide money, is going WAY over the line.
Postal carriers are the only federal employees who have daily contact with almost every American. They definitely deserve to be recognized for the service they provide, especially if it is consistently excellent. But do them a favor and make sure that if you are compelled to give them gifts, the gifts won't get them in trouble. You can call the USPS Ethics Helpline at 202-268-6346 if you have more questions.
Happy Holidays, and thanks to letter carriers everywhere for all they do year-round to deliver for you!!