  We prove that under certain explicit conditions, the Mahler measure of a
three-variable polynomial can be expressed in terms of elliptic curve
$L$-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's
conjecture. In some cases, these dilogarithmic values simplify to Dirichlet
$L$-values. The proof involves a construction of an element in $K_4^{(3)}$ of a
smooth projective curve over a number field. This generalizes a result of
Lal\'in for the polynomial $z + (x+1)(y+1)$. We apply our method to several
other Mahler measure identities conjectured by Boyd and Brunault.
