Two designs of conical-beam array antennas are presented for different fifth-generation applications. They are based on slotted cylindrical waveguides and a travelling-wave topology, where the waveguide is used to progressively excite a cross-slot array. A total of 384 cross-slots, formed by transversal and longitudinal slots, are grouped in rings of eight equally-spaced cross-slots. The propagation of TM01 and TE01 modes in the cylindrical waveguide can provide dual polarization by the independent excitation of transversal and longitudinal slots, respectively. Firstly, a dual linearly-polarized antenna design in the 37-40 GHz band is presented, conforming a high-gain conical-beam pattern. Secondly, a similar antenna design working in the 5G dual-band of 26-30 GHz and 37-40 GHz is also presented. In this last case, transversal and longitudinal slots are designed to radiate at the two different frequency bands of 26-30 GHz (vertical polarization) and 37-40 GHz (horizontal polarization), recently assigned for very high-speed 5G applications. In order to experimentally validate the proposed topologies, the dual-band design has been prototyped by combining 3D-printing and CNC milling techniques, providing high experimental performance. Directive tilted omnidirectional coverages with peak realized gains around 14 dBi have been obtained, as well as a total efficiency between 83% and 90% for both frequency bands.