Or, perhaps ‘sah-na’?
My stepfather was Finnish, from Turku. Since he pronounced it ‘sow-na’ (where ‘sow’ is pronounced like a female pig), that’s the way I say it.
You?
Sow-na. Pronouncing it Saw-na is a good way to get yourself laughed out of the UP.
The predominant ethnic heritage up here is Finnish, so that’s why we stick with the Finn pronounciation.
Speaking of saunas, I better go turn mine on!
“Saw na”.
FWIW, I’m a 2nd generation Floridian (whatever accent that is).
Sweat lodge.
The finnish family up the road from us pronounces it “So - UNE”. Beats me if it’s right.
A link to the article that the OP took the image from. (FTR, without even seeing the bit about the Frazz comic or noticing that the link was from Language Log, I actually just hopped on Language Log and searched for “sauna” to see if I could find anything relevant. Kinda weird.)
Anyway, seems to me the preferred American pronunciation is /ˈsɔːnə/ (/ˈsaːnə/ in California), or “saw-na”. Those of the Nordic persuasion seem to prefer /ˈsaʊnə/ or “sow-na”, but then, we don’t pronounce marriage /mariaʒ/ or “mah-ree-age” with the sound of g in genre, even though that’s the “authentic” French pronunciation.
Of course, as long as the people around you understand you and you don’t get laughed out of town, it’s all /pəˈteɪtəʊ/s and /pəˈtatəʊ/s to me. 
ETA: I don’t mean to trivialize that group of non-Nordic speakers of American English who pronounce it /ˈsaʊnə/ – just trying to remind the pedants not to pick a winner and flame the other side.
Not to be a snot, but neither – for me, it is sah-na. I have a pretty mild (though it can be a bit heavier when I drink) Southern accent, so “saw” and “sow” both have a pronounced “w”, which sauna doesn’t.
I’m sure everyone with the [cot/caught merger](Phonology - Wikipedia history of English low back vowels#Cot-caught_merger) would pronounce it as “sah-na” /ˈsaːnə/ rather than “saw-na” /ˈsɔːnə/. It is rather uncommon IME for such a person to recognize the difference, though–it was news to me when I found out in Linguistics 1 that “cot” didn’t rhyme with “caught” and “Don” didn’t rhyme with “Dawn” in some places.
What’s the UP?
Beat me to it. 
I have never heard it pronounced any way but saw-na.
I thought quite a few people merged towards [ɔ] rather than [a]? Not that the speakers would be aware of the difference.
Nope, it universally goes to [a]. “Cot” sounds like “caught” here, not the other way around.
ETA: Of course, there could be some going the other way. But I’m not familiar with it, and the little research I’ve done into the matter has not turned up any.
What an ambiguous description.
Unfortunately, most people are sorely lacking in linguistics education. I think one semester of linguistics should be required for all college students, personally.
All Finns pronounce it SOW (as in cow) NAH. Only Swedes would say sah-nah.
Johnny LA, if your dad is still alive tell him "terveisia Turusta’ (greetings from Turku). I lived there for about a year in the mid sixties. A very nice town.
I used to be a “saw-na” speaker until I became friends with a UP Finn. He got me saying “sow-na” just out of sheer repetition. Bastard. Now I sound all pretentious, and it’s all his fault! 
Sow-na. I’m an ethnic Finn, grew up speaking Finnish, have also lived and studied in Finland. It hurts my ears and brain to hear the way most Americans pronounce it. Granted, **Fetus **has a point, that words change pronunications when they move into other languages, but so few words from Finnish make it into other languages, it would be nice to keep clean the ones that do.
Both Turkka and my mom are dead. How are your greetings pronounced? the only Finnish I know is ‘Mita kuulu?’ If I were to see the Finnish word for pilot, I’d recognise it. (‘Lentija’?) My mom said the only Finnish she knew was something Turk would say when he was talking to his mom. Sounded like ‘vash-te-meh’. I don’t remember what it means.