Influential is a difficult word. But here goes:

O'Sullivian (the survey aesthetic has been re-appropriated by many contemporary LFCN guys)
Atget (same reason as above)
Evans
Bresson
R. Adams
Shore
Eggleston
Soth

I've defined 'influential' as those photographers who are clearly visually referenced the most in modern work.
I haven't included Ansel or Weston for the same reason I wouldn't include The Beatles in a music list - goes without saying.