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, ..., 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 --> 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.