Neither do I, but it so happens I was on a Linus kick some months ago and took note of whatever tracks I recognized:
Wraparounds (“The Company”):
Happy Go Lucky - George Chase (Major Records) (Some wraparounds feature a “soundalike” of this tune by Hoyt Curtin as well.)
Linus, King of Beasts:
Hicksville - Phil Green (Photoplay Q/Capitol Hi-Q)
Killing Time - Reg Owen (Conroy)
Walking the Plank - Harry Dexter (JW Theme Music, used as the theme for most segments.)
Watch for the Manhole - Ward Sills (Video Moods) (also heard in at least one Rory Raccoon segment.)
Lovable Truly:
Carefree Character - Alan Perry (Conroy) (Used as the main theme for most of these segments.)
Hippety Hop - Franz Mahl (Video Moods)
Idle Fellow - King Palmer (Impress)
Rory Raccoon:
Carefree - Van Phillips (Impress)
Clown’s Dance - Len Stevens (JW Theme Music)
Flippancy - Franz Mahl (Video Moods)
The Happy Hippo - Eric Winstone (Conroy)
Merry as a Grig - Van Phillips (Conroy)
Midget March - Cedric King Palmer (JW Theme Music)
Sports Fever - Ronald Hanmer (JW Theme Music, used as the theme for all Rory segments)
So Hi:
Grotesque - Heinrich Fieschner (JW Theme Music)
Rickshaw Ride - Ronald Binge (Impress)