Georgia - relative vicinity of Atlanta:

Off the top of my head...

- St. Frances de Sales (FSSP) - Mableton, GA
- Our Lady of the Mountains (OF) - Jasper, GA
- St. James (OF) - Madison, GA

Phoenix has some churches to visit for respectable music or dignified liturgy: the Cathedral of SS. Simon and Jude (where @matthewj is director of sacred music), and St. Joan of Arc, where Jeffrey Morse is director of music. There's also an EF Mass at Mater Misericordiae parish, run by the Fraternity of St. Peter.

In addition to those Chonak has mentioned, Phoenix also has an SSPX parish for those so inclined (Our Lady of Sorrows) as well as St Catherine of Siena (EF on MWF and Sunday evening, although I think Missa Cantata is only 1x/mo) and the Newman Center at ASU which has an evening Mass with organ, vernacular propers, and a Latin Ordinary (Missal/Jubilate Deo).

I attended Good Friday this year at St Mary Magdalene in Gilbert and there was a sizable (and young!) choir that sang Gregorian chant and several polyphonic works quite well.

There are several other parishes around with excellent organists and respectable hymnody. Adherence rate to the sacred music ideal varies amongst them.

Also in Gilbert is St Thomas Byzantine Catholic Church, which I was always glad to visit when I lived nearby.

Actually no matter where you are in the Phoenix metro area, there is a Church with sacred music nearby.

Central
The Basilica of St. Mary - regularly has chant, professional choir, pipe organ, very good organist, music at daily Mass Tuesday-Friday at noon
The Cathedral of SS. Simon and Jude - fully Sung Masses, Gregorian propers everywhere (including Spanish Masses), only Gregorian and polyphonic Ordinaries, pipe organ, ad orientem 11am Sundays, 6:30pm Tuesdays, 8:30am Fridays. Music at daily Masses frequently.
St. Thomas the Apostle - choir sings a lot of polyphony, 11am ad orientem

North
St. Joan of Arc - growing program, likely with Choristers in a few years, English chant
St. Bernadette - English and Latin chant frequently

East (Scottsdale)
St. Maria Goretti - semi-professional choir, good motets, English chant, occasionally things from the 1980s/1990s. Pipe organ and excellent organist.

East (Tempe)
All Saints Newman Center (ASU) - 5:00pm Mass, professional men's schola, mostly English chant with some Latin and motets. TLM occasionally.

Far East
St. Mary Magdalene, Gilbert - English chant everywhere, small pipe organ, choristers in formation. Some Masses ad orientem.
St. Anne, Gilbert - 11am Mass has Gregorian chant and motets, fully chanted Mass ad orientem. TLM occasionally.

West
St. Clare of Assisi Surprise - English chant, solid hymnody, sung Masses regularly

South
Mater Misericordiae - FSSP, Gregorian propers, motets, very nice small pipe organ (an ASU practice instrument that was revoiced by Rosales), all TLM
Our Lady of Sorrows - SSPX, all TLM
St. Catherine of Sienna - Hauptwerk organ and chant regularly, all Masses ad orientem. TLM often.

Way out there
St. Margaret Mary, Bullhead City - Pastor is massively supportive of sacred music and probably sings Mass often. Lumen Christi materials in the pews. Choir is probably in a building phase. TLM occasionally.

I've continued to add more parishes to the above list.

It must be fun living in such a rich musical environment. Do the choirs in your area engage in either friendly competitions or collaborative efforts?

It probably looks a lot more impressive than it is - a lot of places are in their infancy of the sacred music movement.

Since Phoenix is so spread out, it's not often that we do much together. We do social events once or twice per year (for CARD CARRYING REVOLUTION MEMBERS) and I usually try to hire a Music Director or two for big Diocesan events that I do the music for (Chrism Mass, Ordinations when held at the Cathedral, etc).

For the most part we just go about our lavish lives as crazed revolutionaries simple workers in the harsh desert wasteland Vineyard of the Lord.

Omaha's Cathedral has an awesome music program.

I didn't answer this, because the OP wasn't asking about my geographical area.
It's kind of a funny question though. Any EF that's not a Low Mass will have "sacred music", though it may not be particularly skillful. I'm assuming we mean "not sacropop". Oddly, I find that churches with the resources to do good music are maddeningly middle of the road, and those with great taste don't always have the resources to pull it off.

Oddly, I find that churches with the resources to do good music are maddeningly middle of the road, and those with great taste don't always have the resources to pull it off.

Isn't that the truth!

Sorry, I hope I'm not too late to post this.
In Miami, FL we have the Cathedral of St. Mary and Church of the Epiphany.
The Cathedral is to the NE and Epiphany to the SW.
There are other parishes that offer a regular EF mass with music that is beautiful, but very different from the other, regularly scheduled masses.
The Cathedral does a lot with a little. Epiphany does a lot with a lot. In my experience these two promote good music while keeping out the less appropriate music.