I always voice the “th” in “with”. I didn’t know there was any other way to pronounce it.
The US norm, in my experience, is to pronounce it unvoiced.
Interesting. And not in my experience. I’m with Ben; I didn’t know it could be unvoiced. Not in my dialects or any place I’ve lived. An upper Midwest thing?
Could well be; we have many remarkable points of interest unique to us, like lime Jello mold “salad” with shredded carrots and riced cream cheese. It shouldn’t even be fed to a dog. I think it’s served at church potlucks as a graphic token that Hell actually exists.
You’ll hear the TH in “with” voiced in some fast delivery or in some combinations, but it’s not the norm; and after all, I suppose we are not best noted for the handsomeness of our speech. Let me clarify: I habitually hear it unvoiced, but maybe it’s all in my head. Now I’m curious. I’ll listen more closely for the truth and get back to you.
Yes, do. If there’s one linguistics truth, it’s that people are notoriously unreliable about self-reporting their own linguistic behaviors (cf. the bit about plosives in that other thread). 
Yeah, I sort of had that thread in mind.
It’s a shame that its apostrophe doesn’t exist while it’s does simply because it’s a chore to sort it’s’ apostrophe from its’ non.
Theirs’ the problem right they’re.
It shouldn’t take too much effort: Remember that “it’s” is “it is”, just the same as “he’s” is “he is”. “Its” is like “his” - we don’t write “hi’s”.
His’n? 
Bob
Man, it’s hard to remember to do that when you’re having a chat. Would there be a risk of skewing the results if I wrote “with” on a piece of paper and asked them to pronounce it?
This is the Heisenberg effect in any kind of fieldwork. When your informants are aware of what you’re investigating, they may change their behavior accordingly. So put your test target in context. Make up three sentences with “with” - one before a vowel, one before a voiced consonant, and one before an unvoiced consonant. Ask them to say each sentence, repeatedly if necessary. But don’t tell them what you’re testing.
Thanks for the guidance. 
But since there will be questions, I can’t just say nothing and expect to get away with it. They’ll think I lost my marbles. If pressed, I wouldn’t even say I was checking on trends in Upper Midwestern speech, because while I have no problem with them still thinking I’ve lost my marbles, Minnesotans love to exaggerate the stereotype at the least provocation, and that won’t do. I’ll have to come up with something as neutral possible. Or I could have them say it first, and then mumble something that passes for an explanation after.
Well, you don’t have to hide that you’re testing pronunciation, as long as the context embeds the element you’re testing.