Can you say 'either' when there are more than two options?

I've heard a lot 'either A or B' in English sentences.

But if there are more than two options, can you put 'either' in the sentences as well?

I mean, 'either A, B or C', 'either A, B, C or D'...
Do they make sense?

e.g. "Swiss people speak either German, French, Italian or Romansh."
(That was an example of 'either A, B, C or D' sentence! Is this grammatically correct??)

Yes, you can definitely say this with the meaning you intend here. I am not sure what some prescriptivist sorts may think of it, but in actual usage this is most definitely allowed in most English dialects (at least those I know of).

It's not the most grammatically correct thing to say, but everyone does it, and there isn't really a good alternative, so you might as well.

I agree with Travis and Uriel: there may be some prescriptivists who oppose it, but it sounds completely natural to me.