^There was also the Leagcy comic series set 100 years after A New Hope, although that did feature cameos by at least Luke, and R2 appeared eventually. But I'm pretty sure the majority of issues didn't feature any characters from the movie.
I personally wouldn't have an issue with getting a completely new crew on the Enterprise or Voyager, but I do have to question whether it would be allowed from a marketing standpoint. I could easily see an argument being made that for a story to be a TNG story it would need to have characters who appeared on TNG.
I'm trying to think how many screen characters are in each current series.
TNG: Picard, Beverly, Worf, Geordi, Data, Lal
DS9: Ro (introduced on TNG, now moved to DS9), Bashir, Quark. I haven't read past the Typhon Pact DS9 stories yet, but I believe more TV characters have come back since then.
VOY: Janeway, Chakotay, Kim, Torres, Paris, The Doctor, Barclay, Vorik,
ENT: Haven't read the Rise of the Federation yet
TTN: Riker, Troi, Tuvok, Pazlar
SCE: Gomez, Duffy, Lense, Stevens
GKN: Klag, Toq, Rodek, Leskit, Kurak