The pronunciation of "Roger That"

No, you just say the "TH" sound.

You can get away with saying "Roger dat." It will just come off like you're trying to be a little silly.

The th sound is a tricky one, and appears all the time in English It’s pronounced by putting your tongue slightly between your teeth, and blowing while vibrating your vocal chords (or just make a sound with your mouth)

It's pronounced just like it's spelled (for the most part, anyway -- I'm sure it's dependent somewhat on dialect). For most native speakers, it's probably not really difficult to pronounce that way.

No, you just say the "TH" sound.

if you struggle with the “th” sound, you will still be understood if you end up pronouncing it with a “d” instead. but we do pronounce it even when spoken quickly

If you find it difficult to say "roger that" quickly, it might be because you're trying to use an unvoiced "th" (as in "thin") instead of a voiced "th" (as in "this"). But I'd have to hear a recording to be sure. If you struggle with this, try putting your tongue between your teeth like you would for an unvoiced "th", but instead of blowing air, just hum. This may not seem like a big difference, but it'd be like saying "foice" instead of "voice" or "soom" instead of "zoom". It matters!

For other instances of this combination, you can practice saying "worthy" or "further". Other words with the same articulation but leaving the th unvoiced would be "birthday", "fourth", "orthodox". There's a body of water in Scotland called the Firth of Forth, which is fun. It is indeed a complicated articulation to make with your tongue, but it nevertheless is a proper voiced th sound immediately following an r.

I would say yes and no. I definitely pronounce the 'th' and not a 'd', but it it somewhat of a hybrid between them when saying that phrase

When I say "Roger that" naturally, as I would say it conversation (especially in its initial setting of two-way radio communication), the th in that is a voiced th, but it's extraordinarily light. It *almost*, but not quite, disappears. If you are having trouble it's probably because you are putting too much emphasis on saying it properly. A native speaker would have learned to say this sound, and then over many years, would have gotten gradually faster, and lazier at making the sound, resulting in something similar to that sizzle you get when you *barely* touch a skillet with a wet finger for a tiny fraction of a second to see if it's getting hot. (Disclaimer: this is a metaphor; don't literally go touching hot skillets. Not responsible for injuries.)

In the case of “Roger that”, as long as you produce the sound by your tongue contacting your teeth (and NOT the roof of your mouth) it will be heard as “correct” even if that produces a pseudo D sound. If you can make your tongue make what you think is a D sound by using your teeth, you’ll be 90% of the way there. In natural speech, it’s not as breathy as most English speakers believe. If any English speaker doubts this, try to say “tanks” but use your teeth instead of the roof of your mouths to make the t sound. You’ll notice, if you’re doing it right, you always say something that would be heard as “thanks”

You need to say both words as they are written, pronouncing every syllable.

I have no difficulty with this at all but then I have a non-rhotic accent. If I try to pronounce it in my poor imitation of an American accent and pronounce the R at the end of Roger, it does become more difficult and feels as though my tongue is getting in the way.

I do pronounce it with the "th" sound. I have never heard anyone say "roger at", but "roger dat" would be expected in dialects with th-stopping.

Try saying it slowly before you worry about saying it super fast. Probably you're speaking too quickly and getting yourself tongue-tied.

No special pronunciation for this. Just do it like it's spelled.  Any unique pronunciations you hear are dialect-specific -- someone who says "dat" instead of "that" in this context would also say "dat" in other contexts. Someone who doesn't use "dat" in other contexts wouldn't use it here either. "Roger 'at," similarly, is just sloppy enunciation. I can definitely see myself pronouncing it this way while speaking quickly, but you'd be able to hear a hint of the "th" if you listened carefully.

You would pronounce it as it is. Roger that. Accents might change how it’s said. A Cajun from south Louisiana might say “Roger dat” simply because that’s how they always say “that” all the time.

“Roger that” is pronounced with a soft/voiceless “th” sound. However, on the radio just say “Roger.” 

I do pre-pronounce the TH, but it's a softer pronunciation than I would use if the word "that" were at the beginning of a sentence.

And ðis is why English should have kept eð and þorn as letters

you could just say 'roger'

There are lots of different accents in English.

I would say [ɻɒ.dʒɨ.ðæʔ] You could substitute [ə] for [ɨ] tho. It's a vaguely centralized reduced vowel that is more high than normal. I might subtly color the schwa a bit to be more r like but only lightly. It would be difficult to fully pronounce that r in an unstressed syllable and then transition into a /ð/. So if you're saying it quickly, the r is the sound most likely to get lost. The th is too important to drop in my accent. note that I frequently code switch between nonrhoticism and rhotacism, but I'd say this the same in either accent.

Any of those options work

It’s Roger That.

"Roger that" and "this or that" are both fully pronounced with the TH.

For discussion not to answer the question: I guess the “r-th” combo is pretty rare in other languages but really common in English: “fourth, Earth, birth, three, ‘other than’, ‘or this’”, etc. How unique is it to English? (I’m remembering learning Russian and getting hung up on “вслух”. American tongues do not like starting a word with “fsl”!)

It might sound more like a stop (like d, t, etc.) , but it’s probably isn’t one. You more so just touch the tongue to the bottom of the top teeth in rapid speech while voicing.

“Rrrrrrroger that”

Typically, it's a term used when talking on the radio such as in aviation where there is some background noise and some of the word might get distorted. So it might not sound the same as hearing someone saying it in person. Maybe that's why you're hearing it that way? But the "th" is pronounced the same as in the word "the."

I just go Roger tha

I find i almost delete the /ð/, or at least do it very lightly.

I do say "Roger that," but I don't emphasize the TH sound or dwell on it. And in fact, it probably would sound more like "Roger dat" if I say it quickly enough. But I do touch my teeth with my tongue like any other TH word.

I know Islanders will pronounce it as "raja dat" but that's besides the point

It is pronounced Rah-djer That. With the th. You can use a d sound for dat, but it's not considered proper.

Make an "S" sound with your tongue at the top of your mouth like you normally would. Keep making it (like a ssssnake) and slide your tongue slowly forward towards your teeth. You should start to hear that "th" sound come through.

Does it help to pronounce the "g" more like a "zh" sound? It won't sound as obvious as changing "that" to "dat"

If you're struggling to pronounce the th, this would sound closer to V than D. Say "roger **v**at" instead of "roger dat".

It’s pronounced Roger that, just like the two words are pronounced individually.

Its roger dat, you will almost never hear the th sound pronounced. Same goes for this, the, then, there and so on. Its spelled a certain way but pronounced more simply. Dat dere, da or de for the and den all are how they truly sound.