Any claim at this point that rock and roll came directly from jazz, I think, would be a minority opinion among people who study these things. One often cited connection between the two genres comes out of the fact that Les Paul was instrumental (no pun intended) in developing the solid-body electric guitar, which also led to electrified bass guitars. Les Paul was a jazz player, although he actually played in a lot of different styles. The book What Was the First Rock 'n Roll Record? by Jim Dawson and Steve Propes pointed to "Jazz At the Philharmonic: Blues Part 2" from 1942 as the first rock record. Specifically, the part of the recording where Les Paul comes in on electric guitar. It's a jazz record, in a pretty recognizable jazz style (Nat King Cole also plays on the full recording).
However, the book Turning Points in Rock and Roll: The Key Events that Affected Popular Music in the Latter Half of the 20th Century by Hank Bordowitz kind of dismisses the claim of that earlier book, while acknowledging the centrality of the introduction of electrified instruments to the new genre: "Even if 'Blues, Pt. 2' wasn't 'the first rock and roll record,' the artists who took up the flag of urban blues, the post-Hank Williams country, even bebop that evolved into rock, took to the electric guitar."
And as that quote alludes to, rock came out of an amalgam of musical traditions. Some of those traditions were played by white musicians, and some of them by black musicians, and rock and roll was the result of a sort of crossover between the two musical communities listening to and being influenced by each other (even if those communities remained largely segregated).
Folk music, blues music, country music, and "Western music", I would argue, were all much more central to what became rock and roll. Jazz played a part, particularly "swing", as did the pop music of the first half of the 20th century in general, but it's hard to find early rock records that really sound anything like a jazz record of the same era or from shortly before. Maybe if you're talking Dixieland and ragtime, you can see some similarities, but if you're talking Duke Ellington-style jazz orchestras or even jazz quartets, it's harder to make the connection.
That said, one of the more direct strains of jazz you do find in early rock is from the likes of Louis Jordan. Chuck Berry said that his desire when coming up was to be like Louis Jordan. Little Richard covered Louis Jordan in his early career. Jordan was a "jazz musician", however, what set him and artists like him apart was that his band eventually started playing in a "rhythm and blues" and "jump blues" style. In other words, if Duke Ellington is on one side of Glenn Miller style big band "jazz" music, then Louis Jordan was very much on the opposite side. By the time rock and roll was being recognized as a unique, new genre, the "rhythm and blues" genre had long had its own Billboard chart, separate from jazz.
And Jordan himself changed a bit over his long career. While his 1949 recording "Saturday Night Fish Fry" is the one most often associated with the "birth of rock and roll", I personally think a stronger connection can be made with his "Beans and Cornbread" of the same year. Compare both to his 1939 recording of "Keep A-Knockin' (But You Can't Come In)", and you can hear a rather different sound. Although, it should be noted Louis Jordan isn't always the best example anyway. He was labeled as "rhythm and blues" in his time, but he was a lot like Ray Charles would later be, in that his recordings could actually be quite diverse. (For example, the B-side to Ray Charles first single, "The Midnight Hour" is a bluesy jazz number, while his first hit, "Mess Around" is jump blues/rhythm and blues/proto-rock, but then he'd record things like "Don't Let the Sun Catch You Crying" which was jazz/pop in the style of Nat King Cole.)
In other words, most of the "jazz" musicians who ended up being directly influential on rock were never playing in a standard "big band" "jazz orchestra" type style, and were often crossing over to other genres. You can chalk that up to that being what rock and roll is, and that is a valid argument. But I think it's important to recognize that most jazz influence came more indirectly, from artists who also played blues, and rhythm and blues and it was the blues aspect that was the more direct link. As an example, Big Joe Turner may have been classified by some as a "jazz" bandleader when he recorded "Roll 'Em Pete in 1938, but he was always a "blues shouter" who played "jump blues", "boogie woogie", and "rhythm and blues", unlike Louis Jordan, who actually did move a bit back and forth between "big band" and "rhythm and blues". Turner's 1954 recording of "Shake, Rattle, and Roll" is often identified with rock and roll, even though it may technically still be "rhythm and blues", but it's not a whole lot unlike the stuff he'd been playing since the 1930s—always much, much further into the "blues" camp than the "jazz" camp, even though he and Louis Jordan ostensibly played in the same "rhythm and blues" genre.
Instead of jazz, as mentioned, I think the more direct through-lines can be heard by most listeners with folk, blues, country, and "Western". Even before rock, these genres started to converge into what was known as "boogie woogie" which quite often sounds like a 1940s/early 50s version of rock, especially once the genre started incorporating electric guitars.
As some examples, you can go back to the 1920s, 30s, and early 40s and hear recordings by folk and blues artists such as the Carter Family (ex: "Single Girl, Married Girl"), Casey Bill Weldon (ex: "You Gotta Do Your Duty"), the Shelton Brothers (ex: "Just Because"), and Leadbelly (ex: "Julie Ann Johnson") and hear recordings that sound quite similar to what would be rock, if they'd just plugged their guitars in and had a backing drummer.
Some artists that early on, in fact, would eventually be adapted into rock and roll. Nirvana's cover of Leadbelly's version of "Where Did You Sleep Last Night" on their Unplugged album is a famous instance. Earlier on, on the Lovin' Spoonful's debut album of 1965, they covered the old blues numbers "Blues In the Bottle" and "Fishin' Blues", among others. Their versions are almost just plugged-in versions of the earlier recordings by Prince Albert Hunt's Texas Ramblers of "Blues In the Bottle" from 1928, and Henry Thomas's "Fishin' Blues" also from 1928, without any major changes. Similarly, the Grateful Dead's "New, New Minglewood Blues" isn't a whole lot different from the original "New Minglewood Blues" by Noah Lewis's Jug Band, just sped up a bit, with a new intro, and played on electric instruments.
Going back further, you can listen to all of Elvis's Sun Records recordings from 1954-55, and they're all covers, mostly just previous "rhythm and blues" songs or "country and western" songs, sometimes with louder electric guitars than the originals, but not always. Arthur "Big Boy" Crudup's original 1946 "rhythm and blues" recording of "That's All Right (Mama)" is probably more raucous than Elvis's version. But Elvis's version of the Shelton Brothers' 1942 hit "Just Because" and his verson of Bill Monroe's 1946 bluegrass standard "Blue Moon of Kentucky" were both sped up and much louder when he played them.
cont'd...