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Several natural and engineered systems may be idealized as sets of spring–mass structures that are coupled in parallel. These range from macroscopic structures such as buildings connected at story-levels with flexible elements, to microscopic systems such as stacked chains of atoms with inter-layer bonds. We realize two possible configurations of such systems by varying the frequency of coupling. One is a configuration where every mass in one chain is coupled to its corresponding mass in the other chain (full coupling). The other configuration consists of chains that are periodically coupled only at certain locations (partial coupling). We develop analytical dispersion relations for both cases, and reveal several interesting and unusual characteristics. Specifically, we show that wave propagation in fully-coupled meta-chains has a dual nature. Depending on problem parameters, the system may behave similar to an acoustic metamaterial (AM) with a locally-resonant band gap or it may allow for the simultaneous propagation of two independent waves. The partially-coupled system exhibits more complex behavior, including nearly flat and negative group velocity bands as well as several Bragg scattering and local resonance band gaps. Furthermore, we present two example devices for wave propagation control using finite prototypes of each meta-chain configuration; realizing a narrow-band pass filter in one of them and a low-frequency rainbow trap filter in the other. Finally,we discuss our findings within the framework of modular design of metamaterial building blocks.