Summary: We construct several numerical approximations for first order Hamilton-Jacobi equations on triangular meshes. We show that, thanks to a filtering procedure, the high order versions are non-oscillatory in the sense of satisfying the maximum principle. The methods are based on the first order Lax-Friedrichs scheme which is improved here adjusting the dissipation term. The resulting first order scheme is \(\varepsilon\)-monotonic (we explain the expression in the paper) and converges to the viscosity solution as \(O(\sqrt {\Delta t} )\) for the \(L^{\infty}\)-norm. The first high order method is directly inspired by the ENO philosophy in the sense where we use the monotonic Lax-Friedrichs Hamiltonian to reconstruct our numerical solutions. The second high order method combines a spatial high order discretization with the classical high order Runge-Kutta algorithm for the time discretization. Numerical experiments are performed for general Hamiltonians and \(L^1\), \(L^2\) and \(L^{\infty}\)-errors with convergence rates calculated in one and two space dimensions show the \(k\)-th order rate when piecewise polynomial of degree \(k\) functions are used, measured in \(L^1\)-norm.