Periodic systems have attracted a significant attention in science and engineering due to the existence of band gaps in their frequency spectra. Here, we study the flexural wave propagation in beams that are periodically connected in parallel and investigate how the contrast in the material and cross-sectional properties may affect the band structure of these systems and their dispersion properties. Results suggest that by changing the mass and stiffness ratios of the two beams, or by changing the inter-beam connection compliance, several band gaps may emerge and that the band gap width, the lowest band gap edge frequency, as well as the nature of attenuation within the gap may be tuned. Furthermore, by considering a hierarchical system of periodically-connected beam elements with different unit cell sizes, we show how the interplay between scales may affect the overall dispersion properties of the system by opening and closing band gaps at different frequencies. These findings suggest that a modular design approach may lead to novel dispersion properties in beam structures. Finally, using a frequency response function approach, we show that the aforementioned results hold in the limit of finite structures.