Most people still have business or some financial responsibilities in the States. So in most cases you will keep your US bank account.
If you are living in Belize, you will want a regular on shore account for general use. Realize that this account is normally denominated in BZ currency and there is a slight currency exchange at some of the banks, while at others, there are not. There are some banks that do have US accounts.
There are also offshore accounts that do not really deal with day to day living, which are more for investors and off shore dealings.
Hope that helps. If you need more specifics let me know.