I'm looking for music similar to the album "Kings of the Wild Frontier" by Adam and the Ants. Specifically, music with "tribal" drumming (especially non-standard drums), vocal chants, and an overall "primitive" or "wild" feel.

Perfect example is the song Kings of the Wild Frontier: http://www.youtube.com/watch?v=xT27A6Sa10k

Another decent, but different example is "Caledonia" by Cromagnon: http://www.youtube.com/watch?v=rPiO_G-DEHs . The pounding drums and raw-ish production really give it a kind of tribal feel.

Finally, a lot of songs by Tool have a kind of tribal vibe, with hypnotic drumming and bass patterns. This also falls under the umbrella of stuff I'm looking for.

I was excited to see this thread; tribal-based music is one of my obsessions, and I was planning to start a thread like this. I'll probably have a lot to recommend, but I'll start with something that fits the more specific side of your brief. Check out Killing Joke's 1983 album "Fire Dances". Much of their material features tribal drumming, but that album in particular has a certain "Adam & The Ants" feel about it.

http://www.youtube.com/watch?v=xZRV_D2nenk
https://www.youtube.com/watch?v=V1p2j04re1g

I will return to this thread later...

Staying with the post-punk scene for a bit, Joy Division and early New Order used a lot of tribal beats, though often in a less wild, more hypnotic way:
http://www.youtube.com/watch?v=j3suV4k7gF0
http://www.youtube.com/watch?v=WU3w0f9NPnk
http://www.youtube.com/watch?v=bmf5gM94m0o
http://www.youtube.com/watch?v=6y52UHWQRQ0

The Cure (Pornography album) may also be of interest:
http://www.youtube.com/watch?v=R_9WSnnBmS0
http://www.youtube.com/watch?v=Yls_lhHdfPc

Other prominent post-punk/proto-goth bands used tribal rhythms prominently, including Bauhaus and Siouxsie. Less obvious bands include:

Ausgang:
http://www.youtube.com/watch?v=jzS0MwBEJcQ
http://www.youtube.com/watch?v=wjz3u3UvYgs

Savage Republic (lots of ethnic percussion!):
http://www.youtube.com/watch?v=wj3mlov8Z4M
https://www.youtube.com/watch?v=RYP2jpJRA2A

Southern Death Cult:
http://www.youtube.com/watch?v=sm9MkkU9FP0
http://www.youtube.com/watch?v=NYgk0UD-bhQ

More to follow, from other genres...

Bow Wow Wow should be right up your alley. The guys from this band were some of the originals from Adam Ant & The Ants before Malcolm McLaren (Sex Pistols' manager) stole them away to construct Bow Wow Wow. http://youtu.be/Zb5Nk0WOF84

Speaking of Sex Pistols, you should also check out John Lydon's post-Sex Pistols outfit called Public Image Limited (aka PiL, Public Image Ltd, etc.). One of their best: http://youtu.be/79gFZNqsTzs

From the realms of space-rock, this classic Hawkwind track (though not really representative of their general sound):
https://www.youtube.com/watch?v=NYpWgV6QJq4

A few bands from UK space/punk scene of the 90s:

Crow
http://www.youtube.com/watch?v=wDzAFEJzQTk
http://www.youtube.com/watch?v=3lg40lwzOiA

Moondragon
http://www.youtube.com/watch?v=92zAMHUldZM

Iowaska
http://www.youtube.com/watch?v=ofGgBodnZLQ

In more industrial/noise-rock territory, hypnotic tribal rhythms were the driving force behind mid to late-era Swans:
http://www.youtube.com/watch?v=fkYDvOjmVl8
http://www.youtube.com/watch?v=ZeTFJfRNgqE

And they in turn were a massive influence on Neurosis:
http://www.youtube.com/watch?v=2DoOWi8mCSk
http://www.youtube.com/watch?v=i9i8T14Eyig

Another of Neurosis' influences were UK crust legends Amebix:
http://www.youtube.com/watch?v=G87ebhiLmOA
http://www.youtube.com/watch?v=DULj4WV2ZcE


Another album I cannot recommend enough is "Ùr", the second album from Swiss tribal/industrial collective Nàda. However, there are no YouTube clips anywhere (I keep planning to upload them myself), so a Google search will find my blog where you can download the album. It's a fantastic album that is a more ethnic/tribal take on "White Light" era Swans, with a bit of Siouxsie in the mix. Highly recommended. The first album "Cèlmetra" is also great, though much more intense and primal.

The Pop Group - Y (1979)
http://www.youtube.com/watch?v=WGNw48HSUPY
a classic! raw, obscure, primitive

Ghost [Japan] - In Stormy Nights (2007)
http://www.youtube.com/watch?v=cZPZw3SlAvI
http://www.youtube.com/watch?v=0Xo1K0wDvQ4
speaking of "Caledonia", they covered it and this album is great

A couple of wild cards. I can trace my love of tribal rhythms back to Mike Oldfield:
http://www.youtube.com/watch?v=bHTy1Wxxf28
http://www.youtube.com/watch?v=4kVcHnP27JI
http://www.youtube.com/watch?v=PycZnSWMvi0#t=2m12s

German industrial goth band Engelsstaub used that Adam Ant drumbeat for their track "Faerieland":
http://www.youtube.com/watch?v=DhksjDQfYnA


And this isn't rock-based at all, but still great - The Ambush, a tribal techno project of Oliver Lieb:
http://www.youtube.com/watch?v=19L1lhN0R-U
http://www.youtube.com/watch?v=GSMujwrdZtA


Finally, a little self-promotion. Here's one of my tracks you might like:
http://www.youtube.com/watch?v=bJ8yXc2jDAY

and I currently play guitar in this band, Jesus Fix:
http://www.youtube.com/watch?v=eFBy_ANt81w

Apologies for dominating this thread, but I've got tons of this shit. Admittedly, I have a habit of lumping anything that features heavy tom beats into the "tribal" category, but the similarities are there.

Here's a few more:

Door Marked Summer (alt/psych band with 2 drummers, evolved into Crow who I posted earlier):
https://www.youtube.com/watch?v=d17lNCLZqYI
http://www.youtube.com/watch?v=RibMzb7MOHk

From the alt-rock scene:

And You Will Know Us By The Trail Of Dead:
http://www.youtube.com/watch?v=jmwq-OJl8T4

Loop:
http://www.youtube.com/watch?v=sbmhoMpoTac

Sonic Youth:
http://www.youtube.com/watch?v=hwpl2n2pbqk

Cay:
http://www.youtube.com/watch?v=z-ojRRNjrMc

Plus a couple more mainstream suggestions:

Midnight Oil:
http://www.youtube.com/watch?v=VhCCSZyFaXA

Paul Simon:
http://www.youtube.com/watch?v=y11yMCnhxZE

Thanks balbulus, I'm currently working my way through all the recs. I thought I'd throw out a few more I know that I didn't mention in my first post:

Tomahawk - Mescal Rite Part 1: http://www.youtube.com/watch?v=u25OLPFqMiE . The whole album's somewhat Native American tribal-y sounding, but this song really manages to capture it.

Jeehun Hwang - Mechwarrior 2 soundtrack. It's a weird mix between tribal, orchestral, and electronic music. All tracks are instrumental and unfortunately very short. Samples: http://www.youtube.com/watch?v=2Fil6W5KGVA and http://www.youtube.com/watch?v=PAQxocc8BMc

Also, thanks noinim, thanks for those 2 recs. I like the beats in Bow Wow Wow, I'll have to see if I can get used to the rest.

Here's a couple more recs, also in the positive punk/post-punk area:
Theatre of Hate:
http://www.youtube.com/watch?v=eX_6K1xGApY

New Model Army
http://www.youtube.com/watch?v=xnZD61tynS4

Vice Squad, this one is from an EP I think, but also from the Shot Away album. A controversial period for the band because the only original member is the drummer. The singer is NOT Beki Bondage:
http://www.youtube.com/watch?v=86CCz_zJngA

Got another one for you: Herne Bay post-punk/noise band Playground. It may be a bit on the raw side, but check it out:

http://www.youtube.com/watch?v=_9imQV6hcpE

I mentioned Nàda in an earlier post; I've finally got round to uploading their 2 albums to YouTube. The 2nd album "Ùr" has a massive tribal/ethnic influence:

http://www.youtube.com/watch?v=qpNJJh2hYpA
http://www.youtube.com/watch?v=-sIfDP_96cY
http://www.youtube.com/watch?v=wGYrGXEzo74
http://www.youtube.com/watch?v=xiMQKL968C4

you might like Les Bâtards Du Nord, though it's just a modern structure, not Rock music.
http://www.youtube.com/watch?v=ySDXn2C5ics