We study sublinear-time algorithms for solving linear systems $Sz = b$, where $S$ is a diagonally dominant matrix, i.e., $|S_{ii}| \geq δ+ \sum_{j \ne i} |S_{ij}|$ for all $i \in [n]$, for some $δ\geq 0$. We present randomized algorithms that, for any $u \in [n]$, return an estimate $z_u$ of $z^*_u$ with additive error $\varepsilon$ or $\varepsilon \lVert z^*\rVert_\infty$, where $z^*$ is some solution to $Sz^* = b$, and the algorithm only needs to read a small portion of the input $S$ and $b$. For example, when the additive error is $\varepsilon$ and assuming $δ>0$, we give an algorithm that runs in time $O\left( \frac{\|b\|_\infty^2 S_{\max}}{δ^3 \varepsilon^2} \log \frac{\| b \|_\infty}{δ\varepsilon} \right)$, where $S_{\max} = \max_{i \in [n]} |S_{ii}|$. We also prove a matching lower bound, showing that the linear dependence on $S_{\max}$ is optimal. Unlike previous sublinear-time algorithms, which apply only to symmetric diagonally dominant matrices with non-negative diagonal entries, our algorithm works for general strictly diagonally dominant matrices ($δ> 0$) and a broader class of non-strictly diagonally dominant matrices $(δ= 0)$. Our approach is based on analyzing a simple probabilistic recurrence satisfied by the solution. As an application, we obtain an improved sublinear-time algorithm for opinion estimation in the Friedkin--Johnsen model.