Songs I often do in song circles, jams, sing-alongs, etc. include
Do, Lord/This Little Light Medley
Vine and Fig Tree (Lo Yisa Goy)
Let There Be Peace On Earth
When The Saints Go Marching In
Just A Closer Walk With Thee
Precious Lord, Take My Hand
He's Got The Whole World In His Hand
Precious Memories
Farther Along
One Day At A Time
Put Your Hand In The Hand
Go Down, Moses
Mary, Don't You Weep
All My Trials
Amazing Grace
Morning Has Broken
This Old House
You Don't Knock
In The Sweet By and By
Abide With Me
Peace Like a River
Guide My Feet
In The Garden
Bashana Haba'ah (Soon The Day Will Arrive)
Hine Ma Tov
Shalom Chaverim
All God's Critters Got A Place In the Choir

If I am singing in a group that would be alienated by mention of Jesus, I stick to the ones that are generally insprirational or mention only "God."
I never have a problem doing the Hebrew songs I mentioned in a Christian group, because there is nothing sectarian about them.

In some jams, though, old gospel songs--e.g., Turn Your Radio On-- are done more or less as historical/musical gems, not as statements of personal belief, so we do any song that has good harmony and good instrumental licks.
If we can have fun singing Banks Of The Ohio, given our elevated collective consciousness today, why can't a Jew, agnostic, humanist, atheist, etc. enjoy singing "I'll Fly Away?" It's a great bluegrass sound!
Genie