Future cellular systems are expected to use millimeter-wave (mm-Wave) frequency bands in addition to the existing microwave bands under 6 GHz. Severe weather conditions, including sand and dust storms (SDSs) and heavy rainfalls, challenge reliable communications over wireless links at those higher frequencies. In such conditions, besides frequency-dependent path-loss, radio signals experience additional attenuation. The SDS attenuation is related to visibility, receiver distance to the storm origin point, soil type, frequency, temperature and humidity. On the other hand, the rainfall attenuation is affected by rainfall rate, polarization, carrier frequency, temperature and raindrop size distribution. Leveraging on experimental measurements carried out in previous works, a novel unified mathematical framework is introduced in this paper to include SDS/rainfall-dependent attenuation in the performance evaluation of terrestrial wireless cellular networks in terms of coverage probability, bit error rate (BER) and achievable rate in the mm-Wave band. Extensive numerical results are presented to show the effects of the different SDS/rainfall parameters on performance, showing that the degradation due to SDS is generally higher than that due to rain and may cause a reduction of even six orders of magnitude in the average achievable bit rate when the frequency increases from 28 to 38 GHz.