Unidirectional/asymmetric transmission of acoustic/elastic waves has recently been realized by linear structures. Research related to unidirectionality of wave propagation has received intense attention due to potentially transformative and unique wave control applications. However, asymmetric transmission performance in existing devices usually occurs only in a narrow frequency band, and the asymmetric frequencies are always within ultrasound range (above 20 kHz). In this work, we design and propose a linear diatomic elastic metamaterial using dual-resonator concept to obtain large asymmetric elastic wave transmission in multiple low frequency bands. All of these frequency bands can be theoretically predicted to realize one-way wave propagation along different directions of transmission. The mechanisms of multiple asymmetric transmission bands are theoretically investigated and numerically verified by both analytical lattice and continuum models. Dynamic responses of the proposed system in the broadband asymmetric transmission bands are explored and analyzed in time and frequency domains. The effect of damping on the asymmetric wave transmission is further discussed. Excellent agreements between theoretical results and numerical verification are obtained.