If they're 0,1 valued, then the following may be what you need. It's call Levy's Borel-Cantelli Lemmas:

Suppose that for natural numbers $n$, $E_n \in F_n$ (a sigma-algebra). Define $Z_n = \sum _{1\leq k \leq n} I_{E_k}$, the number of $E_1,\ldots,E_n$ which occur. Set $e_k = P(E_k | F_{k-1})$, and $Y_n = \sum_{1\leq k \leq n} e(k)$. Then, almost surely,
  (a)  $Y_\infty < \infty$ implies $Z_\infty < \infty$
  (b)  $Y_\infty = \infty$ imples $Z_n / Y_n \rightarrow 1$.

This allows, in essence, for you to use the Borel-Cantelli lemmas even when the variables are dependent, as long as many variables are mostly independent from the earlier ones.

I know it isn't the strong law, but in many circumstances that you'd want a strong law (but don't have it) this suffices.

My reference is "Probability with Martingales", by D. Williams, and this is Theorem 12.15 (with proof) on page 124.