New Dirac points appear when periodic potentials are applied to graphene, and there are many interesting effects near these new Dirac points. Here we investigate the $textit{Zitterbewegung}$ effect of fermions described by a Gaussian wave packet in graphene superlattice near new Dirac points. The $textit{Zitterbewegung}$ near different Dirac points has similar characteristics, while Fermions near new Dirac points have different group velocities in both $x$- and $y$-direction, which causes the different properties of the $textit{Zitterbewegung}$ near new Dirac points. We also investigate the $textit{Zitterbewegung}$ effect influenced by all Dirac points, and get the evolution with changing potential. Our intensive results suggest that graphene superlattice may provide an appropriate system to study $textit{Zitterbewegung}$ effect near new Dirac points experimentally.
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