  We prove a sufficient condition for the vanishing of the modified diagonal
cycle in the Chow group (with $\mathbb{Q}$-coefficients) of the triple product
of a curve over $\mathbb{C}$. We exhibit infinitely many non-hyperelliptic
curves, including the Fricke--Macbeath curve, the Bring curve, and two
1-dimensional families parameterized by certain Hurwitz spaces, for which our
condition is satisfied.
