While my favorite theme songs were the same - Route 66 (my favorite), Peter Gunn, Mr Lucky, Jan Hamer's Miami Vice work - by far and away the best music I heard on TV was from Johnny Staccato.
It was a 1959-60 John Cassavetes project, about a Jazz musician who helped out crime victims on the side. Like Stacy Keach's Mike Hammer did. The entire score was authentic, mostly original Jazz. You didn't have to wait for the credits; musicians like Barney Kessel, Shelly Manne, Pete Candoli and Red Mitchell appeared on camera as characters, playing right in front of you.
The show was short lived; it was a terrific drama for a half hour show and was not cute or funny like Peter Gunn and Mr Lucky were. Cassavetes himself was a very famous indie filmmaker before there was such a thing in the USA. And needless to say, a devoted fan of heavy Jazz. America was not ready for weekly TV this serious in 1959. I was glued to every episode, already hooked on serious drama and music at twelve years of age.
Just because I was a fan of the shows, I fondly remember the theme songs of Sea Hunt and Science Fiction Theater. I do remember that the UK series by Sapphire Films had good theme music. Think The Four Just Men, The Buccaneers, and Adventures of Robin Hood. Very entertaining shows, too. Luckily for me, syndicated and seen on mostly WGN in Chicago.
Then there was Rawhide. We all know the theme, although at this point I picture the Blues Brothers using it as filler in the club gig. Curiously, Sheb Wooley was Pete Nolan, the scout for the cattle drive. His other fame was making the hit record "Purple People Eater." A Country musician and songwriter, his wife's cousin was Roger Miller, whom he taught guitar to. Wooley penned the Hee Haw theme.