“Admit me into your inner chamber!” cries a big, bearded man on the stage of the Tel Aviv rock club Zappa. But Shai Tsabari’s longing isn’t focused on some elusive human lover—he’s talking about God. In the audience, secular young men with tattoos and religious young women in modest kerchiefs close their eyes and sway together, as if Zappa were a synagogue.

Mr. Tsabari is part of a growing movement of Israeli rock musicians who are turning to Judaism for inspiration, fusing tradition with contemporary Israel to find a voice...