For many, but not all, secular homeschoolers "secular" does not mean "without religion". There are two ways to approach this. Some choose to include spirituality and religion in their lives and certain aspects of their homeschooling. For others religion is purely an academic subject, often tied to history or literature. Both groups need alternatives to the usual Bible only approach.
Many of these recommendations would take the place of Bible Stories not the Bible itself. Many of the others are biographies. My children are partial to biographies written by Demi (at least for grammar stage).
What is God? by Etan Boritzer, illustrated by Robbie Marantz, Firefly Books, 1990.
*Religions of the World: With Internet Links by Kirsteen Rogers and Clare Hickman, Usborne, 2005.
*Usborne Encyclopedia of World Religions: Intenet Linked by Susan Meredith, Clare Hickman, Kirsteen Rogers, and Joanne Kirby with various illustrators, Usborne, 2006.
*Note that I am not sure what the difference is between these two books. They seem quite similar to me but I still need to check with someone I know who sells Usborne books. This is one of my pet peeves with Usborne. They sell too many books that are virtually the same thing only with slightly different titles and extra authors. I have Religions of the World and quite like it for the elementary years.
I am not going to list them out but the Eyewitness series from DK does have a book for each of the major religions.
Stories for Children by Isaac Bashevis, Farrar, Straus and Giroux Pub., 1985.
Menorahs, Mezuzas and Other Jewish Symbols by Miriam Chaikin, illustrated by Ericka Weihs, Clarion Books, 1990.
Jakata Tales (animal stories of the Buddha)
Buddha by Demi, Holt & Co., 1996.
Buddha Stories by Demi, Holt & Co., 1997.
The Dalai Lama: Foreward by His Holiness The Dalai Lama by Demi, Holt & Co., 1998.
Stories That Jesus Told: The Parables Retold for Children by Patricia St. John, illustrated by Tony Morris, Morehouse Publishing, 1993.
The Legend of St. Nicholas by Demi, McElderry, 1993.
365 Bible Stories and Verses retold by Muriel Granger, Golden Press, 1974.
*note that I do not recommend Demi's book Jesus. It is about the only book she has written that my family does not like. It is just verses from the King James Bible.
Names Upon the Harp: Irish Myths and Legends by Marie Heaney, illustrated by PJ Lynch, Levine Books, 2000.
Tales from the Mabinogion by Gwyn Thomas and Kevin Crossley-Holland, Overlook TP, 1996.
Autumn Equinox: Celebrating the Great Harvest by Ellen Jackson, illustrated by Jan Davey Ellis, Millbrook Press, 2003.
The Winter Solstice by Ellen Jackson, illustrated by Jan Davey Ellis, Millbrook Press, 1997.
Spring Equinox: Celebrating the Greening of the Earth by Ellen Jackson, illustrated by Jan Davey Ellis, Millbrook Press, 2003.
Summer Solstice by Ellen Jackson, illustrated by Jan Davey Ellis, Millbrook Press, 2003.
Ancient Celtic Festivals and How We Celebrate Them Today by Clare Walker Leslie and Frank E. Gerace, Inner Traditions, 2000.
Muhammad by Demi, McElderry, 2003
Muslim Child: Understanding Islam through Stories and Poems by Rukhsana Khan, illustrated by Patty Gallinger, sidebars by Irfan Alli, Scholastic, 1999.
American Indian Myths and Legends by Richard Erdoes and Alfonso Ortiz, Pantheon, 1985.
The Four Hills of Life: Ojibway Wisdom by Thomas Peacock and Marlene Wisuri, Afton Historical Society Press, 2006.