Which American dialect pronounces "heard" as "hu-yd"?

Short answer: Brooklyn English and New Orleans English.

Longer answer: at present, virtually no still-spoken varieties of American English feature this merger, but in the past, it was common in some accents in New York (stereotypically in Brooklynese) and in New Orleans English.

What we hear in Creedence Clearwater Revival's version of I Heard It Through the Grapevine is called the curl-coil merger.

Here's some of what the Wikipedia article has to say about this merger:

In some cases, particularly in New York City, the NURSE sound gliding from a schwa upwards even led to a phonemic merger of the vowel classes associated with the General American phonemes /ɔɪ/ as in CHOICE with the /ɜr/ of NURSE; thus, words like coil and curl, as well as voice and verse, were homophones...The merger is responsible for the "Brooklynese" stereotypes of bird sounding like boid and thirty-third sounding like toity-toid.

https://en.wikipedia.org/wiki/Rhoticity_in_English#Coil%E2%80%93curl_merger

This feature of the Brooklyn accent is, let's say, old-fashioned. According to the same article, virtually no native New Yorkers born after 1950 speak this way. This feature of Brooklynese remains familiar to younger people because of cultural artifacts from the early 20th century, such as The Three Stooges or Bugs Bunny.*

But John Fogerty, the lead singer of CCR, is not singing with a Brooklyn accent. It's more of a New Orleans accent, as would befit the singer of Born on the Bayou.

That same Wikipedia article explains that, although most American Southern accents are "non-rhotic" (they don't pronounce /r/ after a vowel, unless there's another vowel sound after it), they do not feature the coil-curl merger. An exception was the New Orleans accent.

However, as in Brooklyn, this merger is dying out. Old blues and (Black) rock and roll singers that Fogerty admired may have spoken this way, but very few people in New Orleans (or anywhere) speak like this today.

*According to Bugs Bunny's voice actor, Mel Blanc, Bugs actually has a Flatbush accent, which is equal parts Brooklyn and Bronx (https://walkoffame.com/bugs-bunny/).

You asked for the 'geographic region' for the word 'heard' when pronounced to rhyme with void, as in the CCR John Fogerty version of I Heard It Through the Grapevine. Many answers gave you phonetic explanations. But for the geography, the 'sound' you are hearing is New Orleans. Fogerty modeled many of his themes as well as his lyrics, including pronunciation, after the images and street patois of that famous area. Born on the Bayou, New Orleans, and others. The ties to that city and area are so noticeable that a very frequent question about Fogerty and the boys is, "aren't they from Louisiana?' Quora has an entire post on that subject! https://www.quora.com/How-come-Creedence-Clearwater-Revival-whose-members-are-all-from-California-spills-its-love-to-Louisiana-in-every-other-song.

Hopes this helps.

Except for those with rhotacism (inability to roll R), pretty much all American accents pronounce the r sound in what linguists call the "NURSE" set of words (heard,worse,perverted,girl,world,etc) even if the accent is otherwise non-rhotic. The dropping of r in the NURSE set in North American English is really limited to those natives of Brooklyn, New Orleans and Charleton SC that are a minimum of 80 years old.

Fogerty SINGS about working class Louisiana-themed ("swamp") stories, (low income black or Cajun Southerners offering the trope of the proud but tired endurers of simpler, more "original" types of soul challenging lives, as opposed to the trope of the "knowing", complex, and high speed challenges of the Northeast urban poor). But the accent Foggerty uses to build his art, is NOT Louisiana as suggested, lyric content notwithstanding. It's very definitively the post war New Jersey ("Nuh Joyzee") working class accent adopted by many U.S. singers of the contemporary counter culture, to convey (whatever their privileged suburban Anglo white middle class background may factually have been) that they were sage fellow travelers rejecting the values of privilege and capitalistic greed held by their parents, and speaking for (and with the street cred of) the more "authentic" working class (as conveyed by using the (then) highly recognizable accent of the "unvarnished" low income folks of Jersey.

Martha Stewart, who hails from Jersey herself, worked very hard to rid herself of this accent exactly because of those very same "working class" connotations. Not consistent with the image of expert sophistication she wanted to convey. In those days, your accent, whether in England or U.S., pre-identified you (fairly or not) as belonging to a particular socio-economic demographic.

Historically, in the immediate pre- and post-war periods, Jews and Italians were the predominant immigrants to New York and New Jersey. Stereotypically, Italians were known for manual labour e.g. construction, Jews for more esoteric occupations (including studying the respective roles of man and God, and their responsibilities to each other as a form of discourse rather than a set and unexamined doctrine). A frame of view to which the counter culture and their song writers were drawn in their quest to redraw the capitalistic, war and depression causing culture of those "over thirty", and to create a new social and spiritual contract in the world.

Many of the best song writers of the CCR era (Dylan, Diamond among them) came from this Jewish tradition of discourse-based role exploration. Even Karl Marx, idol of the counter-culture, who later in his life preached atheism and anticapitalism, was born into the Judaic tradition of examining the human role and responsibility in established order. Many in the counter culture saw themselves as the proponent of the conscientious "common man". The easily recognized Jersey accent provided the idiom through which they could convey their conscience-raising lyrics.

The New York accent of the time was inflected with a combination of Italian, Jewish, and the already establish Black community, giving it an urban sophistication, while in the New Jersey accent, the Jewish (Yiddish) phonemes are more prominent, and (to the ears of those born in the post war era at least) more easily (read "stereotypically") recognizable as emblematic of the simple burden carrying working class in whose "voice" singers of the time proposed to speak. It therefore works better as a tool to telegraph the street cred. of the of songs about the travails of the "common man" on whom RFK and the counter culture artists were focusing.

Interest in Flatbush and Bronx accents would come in music of later decades, to deliver the more jaded, post-counter-culture tales of the next era's singer-lyricists. (e.g. Lou Reed).

On top of this, the British (musical) invasion of "blue-eyed soul" and psychedelia (e.g. Stones, Animals, late Beatles, Donovan, Led Zep) had the softened "r's" you speak of, which were also adopted by many U.S. artists of the 60's and 70's, cos they made the words sound richer, more poetic, and time transcending. (The softened, open ended "r" lets you hold the note longer, and even wail into it, affording more room for depth and emotion).

The John Fogerty singing accent is therefore a combination of New Jersey working class Jewish/Yiddish - a badge of the counter culture indicating affinity with the "authentic" contemplative common man, turning their back on middle class capitalistic values, and looking for deeper more poetic realities - and the softened "r's" of the British invasion's blue eyed soul / psychedelic rock, complementing the sounds of that deeper and more poetic search.

If you lived at this time, grooved to this music, and were part of a garage band with aspirations to hit fame, you practiced these accents to get the sound right (esp the softened "r's": outside the U.S. the Jersey thing less so - though the English liked to take on what they thought were U.S. accents in their songs. Listen carefully --- that's not an English accent the Stones are singing in).

It's true that the French basis of the Cajun accent has softened "r's" as well, but they differ from the British ones. I've never heard a Cajun speak in a "joyzee" accent. Joyzee is more nasal and to the front of the mouth, with broadly pronounced vowels. Cajun comes from further back and deeper in the throat, and the vowels are much less broad. (pronounce the word "garcon", and you might see what I mean. The lips close around the "on" sound). A softened "r" is about the only thing Cajun and Jersey accents have in common. You would never mistake Troy Landry's or Russel Honore's classic Louisiana accents with John Fogerty's vocalizations.

I recall it was a topic of the time (60's and 70's, when I worked for Capitol Records) to query how the beloved CCR came to sing swamp rock with Jersey-Jewish/British accents. My favorite answer: who cares, it sounds perfect doesn't it?