Run, DMC and Jam Master Jay.
Sick Puppies have only three. They’re a metal-ish band out of (I believe) Australia. One of them is a chick too.
Minutemen
Shonen Knife
Beastie Boys
Föllakzoid
Zeke
UFOmammut
When Gabriel left, the band opted to continue on with the remaining four members. After two more albums, Steve Hackett left, …And Then There Were Three…
The Three Degrees
Fun boy three
Thompson twins - from around 1982 onward
Rory Gallagher - his band was a 3 piece outfit
The Nice (Keith Emerson, Brian Davison and Lee Jackson)
Refugee (Patrick Moraz, Brian Davison and Lee Jackson)
Tin Machine (David Bowie and the Sales Brothers)
Zebra
America
The mid Seventies edition of King Crimson (John Wetton, Robert Fripp and Bill Bruford)
Dixie Chicks
UK (John Wetton, Terry Bozzio and Eddie Jobson)
Hamilton, Joe Frank, and Reynolds
Guitar Wolf!
100 Flowers (Much better known as The Urinals, but I think there was a fourth member by then)
Add N to (X) (although this particular song features Alison Goldfrapp guesting on vocals)
Oneida (originally a quartet, but one of them quit)
Yo La Tengo
Oblivians
Bad Sports
Mean Jeans
The Trashwomen
Beat Happening
Best Friends Forever
Le Tigre
Black Panties
Black Time
You are the first person I have ever encountered IRL in any way that had ever heard of this/these bands besides me. That totally fucking rocks!
Cool!
Instead of paying off my college loans, I bought records. Lots and lots of them. I’ve been flipping through my collection looking for more three-pieces, and there are way more than I would have thought. I’m only up to the end of the “B’s”, and I should probably stop, or I’ll be here all day.
Ben Folds Five was the same way. They were a really tight band for a minute, and managed to put out 3 albums with more than just Ben doing the stuff (still he was doing the bulk of the stuff). But once the band broke up it turns out he didn’t need those guys at all. On his first solo album he played all the instruments and it really just sounded like a BFF album.
Johnny Cash and the Tennessee Two, before they hired W. S. Holland to play drums.
The Jam,
Weller, Buckler, Foxton
Not sure how well known they are in the states but in the UK they were massive from 1976-82, right up there with such as The Clash.
“Aren’t they a quartet? Hamilton, Joe, Frank, and Reynolds?”
“I always thought they were a duo. Hamilton Joe Frank, and Reynolds.”
“No, they were a trio. Hamilton Joe, Frank, and Reynolds.”
“Wait! They were a quintet! Hamilton, Joe, Frank, Ann, Reynolds.”
Beck, Bogert & Appice
Medeski, Martin, and Wood
Animal Logic (though they had side support)
Throwing Muses became a trio after Tanya Donnelly left to form Belly
Giles, Giles, and Fripp
Peter, Paul, and Mary
the Kingston Trio
Hot Tuna are usually a duo but occasionally add a third member. They usually include said short-term member as a full member, not a side musician.
Trio- a German band. They had three members, presumably because Germans don’t do irony.
Tangerine Dream
Oh, and …
Dirty Three
Tex, Don and Charlie
Los Trios Ringbarkus
Another brother act: Los Lonely Boys: Henry, Jojo, and Ringo Garza.
Was going to mention them. They rock.