The easiest condition would be a bound on $\sup_i \mathbb{E} X_i^6$, which would allow you to apply the [Berry–Esseen theorem][1].  More generally, if for some $0<\varepsilon < 2$ you have a uniform bound on $\mathbb{E} |X_i|^{4+\varepsilon}$, then you can apply the [Lyapunov condition][2] (which, as I noted above, follows from Lindeberg's condition by the Markov/Chebyshev inequality).


  [1]: http://en.wikipedia.org/wiki/Berry-Esseen_theorem
  [2]: http://en.wikipedia.org/wiki/Central_limit_theorem#Lyapunov_CLT