Sounds odd. The only time 'nor' sounds fine to me is in the neither...nor construction.
In spoken (American) English I would expect to hear: "I don't work on Saturdays or Sundays."

I changed it to include the "neither".
Would the following be correct as well?
"I don't work neither on Saturday nor Sunday."

It's still wrong. The way it is now there is a double negative (don't - neither). If you want to use neither you have to make the verb positive.
"I work neither Saturday nor Sunday."
I can't figure out what to do with the 'on'. It just sounds wrong mixed with neither. But if you were to use it after neither, you'd have to put it after nor as well. But like I said, it just doesn't sound natural with on.

Or "I work neither on Saturday nor on Sunday".

I resorted to CK's first suggestion. Thank you all.

You could have left them; quite often, the history and discussion a sentence has is more useful than the sentence itself.

Yes, I'd prefer comments to stay, as well. Then other might not suggest the same thing again (or the contrary, if you already explained it well). Please don't delete comments, even if they're only "Please add a comma!" or so.

By the way, I seem to misunderstand this sentence. "Don't" plus "either...or" to me sounds like he is free on either Saturday or Sunday, but he isn't sure when exactly.
Is the sentence really equal to "I don't work on both Saturday and Sunday." and *not* to "I am free on either Saturday or Sunday."?
I'm not a native speaker, though.

Why? On the contrary, please quit deleting your comments. They're very valuable for the sentences.

Vortarulo,
The ambiguity doesn't exist for me. To get the meaning you're asking about, I'd say something like: "I either don't work Saturday, or I don't work Sunday."

Alright. Then it's just my non-native intuition. :)