This paper presents a new design for a dual-band double-cylinder dielectric resonator antenna (CDRA) capable of efficient operation in microwave and mm-wave frequencies for 5G applications. The novelty of this design lies in the antenna’s capability to suppress harmonics and higher-order modes, resulting in a significant improvement in antenna performance. Additionally, both resonators are made of dielectric materials with different relative permittivities. The design procedure involves the utilization of a larger cylinder-shaped dielectric resonator (D1), which is fed by a vertically mounted copper microstrip securely attached to its outer surface. An air gap is created at the bottom of (D1), and a smaller CDRA (D2) is inserted inside this gap, with its exit facilitated by a coupling aperture slot etched on the ground plane. Furthermore, a low-pass filter (LPF) is added to the feeding line of D1 to eliminate undesirable harmonics in the mm-wave band. The larger CDRA (D1) with a relative permittivity of 6 resonates at 2.4 GHz, achieving a realized gain of 6.7 dBi. On the other hand, the smaller CDRA (D2) with a relative permittivity of 12 resonates at a frequency of 28 GHz, reaching a realized gain of 15.2 dBi. The dimensions of each dielectric resonator can be independently manipulated to control the two frequency bands. The antenna exhibits excellent isolation between its ports, with scattering parameters (S12) and (S21) falling below −72/−46 dBi at the microwave and mm-wave frequencies, respectively, and not exceeding −35 dBi for the entire frequency band. The experimental results of the proposed antenna’s prototype closely align with the simulated results, validating the design’s effectiveness. Overall, this antenna design is well-suited for 5G applications, offering the advantages of dual-band operation, harmonic suppression, frequency band versatility, and high isolation between ports.