In general, television Westerns seemed to have very good theme songs.

The High Chaparral, composed by David Rose:
https://www.youtube.com/watch?v=dEOGhUBgtAQ

Bonanza, by Jay Livingston and Ray Evans:
https://www.youtube.com/watch?v=8xi172QhAPA

Rawhide, composed by Dimitri Tiomkin and Ned Washington, and sung by Frankie Laine:
https://www.youtube.com/watch?v=3_rHrLHIeyI

The Virginian, composed by Percy Faith:
https://www.youtube.com/watch?v=hjjUlljGNK0

Lawman, composed by Jerry Livingston and Mack David:
https://www.youtube.com/watch?v=I4DPIMZcVSs

The Ballad of Paladin (theme from Have Gun Will Travel), composed by Johnny Western, Richard Boone and Sam Rolfe, and sung by Western:
https://www.youtube.com/watch?v=8RdXs8xnfyw

Old Trails (theme from Gunsmoke), composed by Rex Koury:
https://www.youtube.com/watch?v=h5E-6AJuqjI

Theme from The Rifleman, composed by Herschel Burke Gilbert:
https://www.youtube.com/watch?v=OTu47r5R5Mo

Wagon Train (second theme), composed by Jerome Moross:
https://www.youtube.com/watch?v=BESu7Ly18uY