KEITH URBAN went under the knife this past Tuesday, and had that throat surgery he's been talking about.  And on Friday, his spokesman said Keith was doing fine, and he'll be able to talk and sing again in three weeks.

Keith reassured his fans before the procedure that it's a pretty common thing if you sing and tour hard . . . which obviously he does.  It just means you've hemorrhaged a vocal cord, have to repair the damage, and take a break.

 The day before the operation, he posted a video thanking fans for their support.  Quote:  "I want to thank you for sending such good wishes for my upcoming surgery.  I can't express enough my gratitude to you guys; it feels like family."

People sent him cards, gifts, emails, flowers . . . and hand-written notes to hold up when he needs something and can't talk.