Shelly I am not too familiar with the use of "unglazed" tile on counter tops. I believe almost all the ceramics and porceleins are fired in a way that they are glazed or sealed. It is important that any tile used on a counter is not pourous because it could stain and harbor bacteria. That doesn't mean it has to have high gloss. I believe you can find very attractive matt finishes such as the porcelein faux tumbled marble, but in reality that is a sealed tile.
Some of the natural stone tiles are of course not glazed, but can be sealed. I think any tile or stone that you are attracted to could be serviceable provided you seal the surface and grout. There are so many selections out there. Hope you find one you like.