Aside from Durin, the names of the seven fathers are not known. This is noted in "Of Dwarves and Men" (published in History of Middle-earth 12):
But of these ancient times only one name was in the Third Age preserved: Durin, the name they gave to the prime ancestor of the Longbeards and by which he was known to Elves and Men. (It appears to have been simply a word for 'king' in the language of the Men of the North of the Second Age.)
The names of the clans are however known, and are also recorded in "Of Dwarves and Men":
In the Dwarvish traditions of the Third Age the names of the places where each of the Seven Ancestors had 'awakened' were remembered.
- The Firebeards: in the Blue mountains
- The Broadbeams: in the Blue mountains
- The Longbeards: Mount Gundabad
- The Ironfists: somewhere in the East
- The Stiffbeards: somewhere in the East
- The Blacklocks: somewhere in the East
- The Stonefoots: somewhere in the East
Although there were seven clans there were only four awakening places: the Blue Mountains, Gundabad, and two in the East (unnamed, but presumably in the Red Mountains).
All of these clans were still around during the Third Age, and were in communication with each other; again from the same source:
Though these four points were far sundered the Dwarves of different kindreds were in communication, and in the early ages often held assemblies of delegates at Mount Gundabad. In times of great need even the most distant would send help to any of their people; as was the case in the great War against the Orks (Third Age 2793 to 2799).