We prove cohomological vanishing criteria for the Ceresa cycle of a curve C embedded in its Jacobian J: (A) if $\mathrm{H}^3(J)^{\mathrm{Aut}(C)} = 0$, then the Ceresa cycle is torsion modulo rational equivalence; (B) if $\mathrm{H}^0(J, \Omega_J^3)^{\mathrm{Aut}(C)} = 0$, then the Ceresa cycle is torsion modulo algebraic equivalence, with criterion (B) conditional on the Hodge conjecture. We then use these criteria to study the simplest family of curves where (B) holds but (A) does not, namely the family of Picard curves $C \colon y^3 = x^4 + ax^2 + bx + c$. Criterion (B) and work of Schoen combine to show that the Ceresa cycle of a Picard curve is torsion in the Griffiths group. We furthermore determine exactly when it is torsion in the Chow group. As a byproduct, we deduce that there exist one-parameter families of plane quartic curves with torsion Ceresa Chow class; that the torsion locus in $\mathcal{M}_3$ of the Ceresa Chow class contains infinitely many components; and that the order of a torsion Ceresa Chow class of a Picard curve over a number field K is bounded, with the bound depending only on $[K\colon \mathbb{Q}]$. Finally, we determine which automorphism group strata are contained in the vanishing locus of the universal Ceresa cycle over $\mathcal{M}_3$.