  In this article we develop a new method for relating Mahler measures of
three-variable polynomials that define elliptic modular surfaces to L-values of
modular forms. Using an idea of Deninger, we express the Mahler measure as a
Deligne period of the surface and then apply the first author's extension of
the Rogers-Zudilin method to Kuga-Sato varieties to arrive at an L-value.
