Although it surely accomplishes nothing more of substance than more elementary-sounding arguments, taking Fourier transforms in the sense of tempered distributions gives a quick outcome: the Fourier transform of $e^{i\xi x}$ is a constant multiple of Dirac $\delta$ at $\xi$, so the Fourier transform of an $L$-periodic functions is supported on (edit: thx @Yoav Kallus) the dual lattice $\check{L}$, as tempered distribution. Thus, for $f_i$ $L_i$-periodic, $0=\sum_i f_i$ gives $0=\sum_i \widehat{f}_i$ as tempered distribution, and we look at supports. Depending what hypotheses you really want/need on the lattices, we easily reach conclusions: for example, if $L_i\cap L_j=\{0\}$ for any two $i\not=j$, (edit) then the same is true of the duals and the support of every $\widehat{f}_i$ appearing must be (contained in) $\{0\}$.
The pairwise intersections can be intermediate discrete subgroups of $\mathbb R^n$, of course, without being lattices or $\{0\}$.