One was cowritten by Glen Moore and Milton Subotsky, the same guys who wrote "Rock Therapy" and "Lonesome Train (On A Lonesome Track)". Another was composed by Peppermint Harris (who wrote "Raining In My Heart" and "I Got Loaded"). It was recorded by Amos Milburn. A third was written and recorded by Richard Berry (author of "Louie Louie" and "Have Love, Will Travel"). I found a fourth on a 1962 single by The Blue Jays (a quartet from LA).
Crab Devil reports that "The most famous rendition of the Moore-Subotsky song would have to be that by Jimmy Cavello and His House Rockers, a recording featured prominently in the 1956 film titled, um, Rock Rock Rock. For some reason, the Brits of the period seem to have gone nuts over that song, which was covered in (presumably) rapid succession by Art Baxter, Don Lang, Shorty Mitchell, and Tony Crombie."
The announcer who introduces Lee Young (with Woolf Phillips And His Skyrockets) calls their "Rock Rock Rock" (1953) "Britain's first-ever rock song." Crab Devil also discovered "a wildass acoustic rockabilly thrashout obscuro" demo from 1956 by Eddie McCall, and a 1959 single by Max Alexander and the Hi-Fi Combo which "happens to be identical to a 1971 release (or I suppose reissue) attributed to Joe Gene and the Cordells and titled 'Rock Everybody Rock'." (Both singles have the same song on the flip side, "Little Rome".)
An R&B group from Philly called The Big Boys The Big Boys released a 1955 single called "Rock-Rock-Rock-A-Bye Baby", and the b-side of Dusty Boyd & The Rockers' 1958 single is called "Rock, Rock, Rocket Ship". Crab Devil praises its "garage-caliber rock 'n' roll with trashcan drumming, echo-laden guitar, overwrought hillbilly vocals, and mystifying space-age lyrics."
Another "Rock Rock Rock" b-side is from a 1963 single by The Dixiebelles. It was written by Bill Justis, and includes the immortal couplet, "Yeah, rock rock rock! / Yeah, put it in your sock!" Crab Devil and I found a few more from later decades to fill out this set.
Papa John Phillips wrote a "Ride Ride Ride" for his early 60's combo The Smoothies, and Jeff Barry wrote one for The Archies. Lynn Anderson's momma wrote one for Lynn to sing (I prefer Sleepy LaBeef's version). Bob Crewe cowrote one for The Eleventh Hour (a studio group that also recorded the first version of "Lady Marmalade"). There's even a "Ride Ride Ride" by a Scientology swing band. Put that in your sock.