W. Burwell - Sporting Youth (Synchro-Fox) (Used as the sports theme of the show)
Ronald Binge - Man In A Hurry (Impress) (Used in Sports-O-Phone 07/02/59)
Wilfred Burns - Cathedral Ceremony (Conroy) (Used in Price The Guest 07/06/59)
King Palmer - Monkey Tricks (Paxton) (Used in Lawrence Fletchenberger 07/07/59)
King Palmer - Rollicking Rhythm (Paxton) (Used in At Home With The Voting Murchinsons 07/09/59)
Van Phillips - Good Clean Fun (Impress) (Used in Grand Motel 07/14/59, Let's Take A Tour 01/11/60 and Biff Burns Sports Room 02/15/60)
Trevor Duncan - Synchro Stings (Impress) (Used in Grand Motel 07/14/59 and The Adventures Of Sherlock Sage 08/06/59 )
Steve Race - East Side West Side (Impress) (Used in Gumshoe Flattery 08/03/59 and The Place For Heroes 12/30/59)
Steve Race - Urgent (Impress) (Used in Gumshoe Flattery 08/03/59)
Frank Ventre - Coliseum March (Synchro-Fox) (Used in The Adventures Of Sherlock Sage 08/06/59 and Arthur Sharp From Duluth 08/24/59)
Norman Whiteley - Signing On (Impress) (Used in The After Hours Television Show 08/11/59)
Paul Meyer - Valse Artois (Conroy) (Used in Lawrence Fletchenberger 08/12/59)
Van Phillips - Twinkle Toes (Short Version) (Impress) (Used in Mr. Science 08/14/59 and Jack Headstrong 10/01/59)
Lou DeFrancesco - Intercollegiate March (Synchro-Fox) (Used in Unrehearsed Courtroom Of The Air 09/02/59)
Trevor Duncan - Grip Of The Law - Opening (Impress) (Used in Big Motor Race 09/02/59)
Trevor Duncan - Grip Of The Law - Dramatic Penultimate (Impress) (Used in Big Motor Race 09/02/59)
Trevor Duncan - Grip Of The Law - Closing (Impress) (Used in Big Motor Race 09/02/59)
Kurt Rehfeld - Three Fanfares No. 1 (Impress) (Used in Wally Ballou On Coney Island 09/04/59)
Ronald Binge - Fiesta (Impress) (Used in Crooning - It's Origins And Place In The Modern World 09/15/59)
Larry Simmons - Hold 'Em (Synchro-Fox) (Used in Jack Headstrong 10/01/59)
Steve Race - Conflict (Impress) (Used in Jack Headstrong 10/01/59)
Bruce Campbell - Tempo For Strings (Impress) (Used in Bob And Ray's Trophy Train 10/02/59)
Jack Belasco - All Together (Synchro-Fox) (Used in Professor Graggins' Space Ship 10/26/59, Sports-O-Phone 01/01/60, and Armed Forces Humor 02/11/60)
Van Phillips - Flea Circus - Coach Ride (Impress) (Used in Al "Bow-Wow" Perthwait 12/05/59 and One Fella's Family 12/25/59)
Philippe Pares - Official Occasion (Synchro-Fox) (Used in Army Amateur Hour 12/07/59)
Maurice Grew - Jay Walker (Impress) (Used in Wally Ballou Goes To Sweetwater, Vermont 12/11/59)
Lou DeFrancesco - Ski Galop (Synchro-Fox) (Used in Lawrence Fletchenberger 01/21/60)
Buddy Abinger - High Day (Impress) (Used in a few episodes, but I can't remember what episodes.)
Van Phillips - Flea Circus - Ballerina (Impress) (Used in Webley Webster Is In Charge 01/28/60 and The Green Pickerel 02/05/60)
Bruce Campbell - Sceptered Isle (Impress) (Used in Bold Journeyman 02/04/60)
Maurice Grew - Tittle Tattle (Impress) (Used sometimes as the Mr. Science theme, and sometimes other themes as well in several episodes)
John O'Notes - In Your Arms (Impress) (Used in Barry Campbell Record 07/21/59)
Trevor Duncan - Paroxysms (Conroy) (Used in The Adventures Of Sherlock Sage 08/06/59)
Steve Race - Upward Sweep (Impress) (Used in Arthur Sharp From Duluth 08/24/59)
Van Phillips - By The Mosque (Impress) (Used in Ace Willoughby 07/01/59)
William Davies - Playful Link (Impress) (Used in Setting The Record Straight 10/22/59)
W. Westphal (Frank Westphal) - It's An Old Fashioned Locket (Synchro-Fox) (Used in Dean Archer Armstead Report 09/07/59)
Gilbert Vinter - Sporting Chance (Impress) (Used in Dean Archer Armstead Report 09/07/59)
John Zamecnik - Nydia (Synchro-Fox) (Used in Project Farsight 09/25/59)
Roger Roger - Majestic Title (Synchro-Fox) (Used in The Gourmet Club 11/09/59)
Wolf Droysen - Uneasy Dream (Impress) (Used in Clock Hospital 11/10/59)
Franz Mahl - Broadway Ramble (Video Moods)
John Marion - Sweet Remembrance (Video Moods) (Used as the One Fella's Family theme)
John Marion - Private Eye (Video Moods)
And as a bonus, here are two tracks I found from the "Bob and Ray: The Two and Only" album:
Gilbert Vinter - Sporting Chance (Impress)
Tolchard Evans - Men Of The Mounties (Impress)
If I find more, I'll bump this thread.