I planned in a future playthrough to make a stealth character, with Assassin Class, since I've never played one before in Morrowind, and I want to join the Morag Tong. But I read somewhere that I have to be a Dunmer to be accepted. Is that true?

No, you however... have to find their hidden hideout... its the only place they'll let you join...

Thanks for the reply. I can handle the discovery of the secret location, I was worried that once I got there I couldn't join because I wasn't Dunmer.

I think you were confusing the Morag tong with the Camona Tong (which isn't really joinable even if you are a Dunmer).

Actually I was following http://tamrielchronicles.com/html/mwfactions.html that says at the Morag Tong: Joinable, though only Dunmer accepted. But that was the only place on the internet where I saw this.