Well, since you seem to be defining "60's" music rather loosely (I don't think Aerosmith or Queen officially formed until 1970, though individuals played in the 60's...) I'll toss out a few:
Johnny B. Goode (Chuck Berry)
Lovers who Wander (Dion)
Only the Good Die Young (Billy Joel)
Runaround Sue (Dion)
Steel Bars (Micheal Bolton)
You don't Mess Around With Jim (Jim Croce)
Your Mama Don't Dance (Loggins & Messina)
Chain of Fools (Clint Black & Aretha Franklin version)
Enchanted (Stevie Nicks)
A lot of these songs can also be treated as cha chas, or west coasts, or...you get the idea. This gives some choices for folks who don't like swing! And some, like Croce's song, are just "fun."