Bank of Asheville offers 3.65% up to 25000:
https://www.bankofasheville.com/80-ezchecking/

Asheville Savings Bank offers 3.75% up to $25,000:
http://www.ashevillesavingsbank.com/
It's a personal checking called The Best Checking

Southcoast Community Bank in Charleston, SC offers 3.25% up to $35,000:
http://www.southcoastbank.com/person...-checking.html

All require 10 debits per month and at least one direct deposit or automatic bill pay. Also, these rates are subject to change. In the last three years, these rates have slowly come down (were over 5% 3 years ago).

You can open up accounts at any of these banks online.

I would imagine that there are community banks in other cities that have similar offers. In my experience, interest rates at community banks are always higher than at national banks.