We use the statistics of regions above or below a temperature threshold (excursion sets) to study the cosmic microwave background (CMB) anisotropy in models with primordial non-Gaussianity of the local type. By computing the full-sky spatial distribution and clustering of pixels above/below threshold from a large set of simulated maps with different levels of non-Gaussianity, we find that a positive value of the dimensionless non-linearity parameter fNL enhances the number density of the cold CMB excursion sets along with their clustering strength, and reduces that of the hot ones. We quantify the robustness of this effect, which may be important to discriminate between the simpler Gaussian hypothesis and non-Gaussian scenarios, arising from either non-standard inflation or alternative early-universe models. The clustering of hot and cold pixels exhibits distinct non-Gaussian signatures, particularly at angular scales of about 75 arcmin (i.e. around the Doppler peak), which increase linearly with fNL. Moreover, the clustering changes strongly as a function of the smoothing angle. We propose several statistical tests to maximize the detection of a local primordial non-Gaussian signal, and provide some theoretical insights within this framework, including an optimal selection of the threshold level. We also describe a procedure which aims at minimizing the cosmic variance effect, the main limit within this statistical framework.