Boson stars are stationary, axially symmetric solutions of a complex scalar field theory coupled to gravity. Recently, multisolitonic configurations interpreted as static chains of multiple boson stars bound by gravity and carrying no angular momentum were reported. We propose to generalize those solutions to the stationary case by constructing chains of rotating boson stars and analyze their properties. The nonlinear elliptic field equations are solved using the finite element method. We find that chains with an even number of constituents exhibit the same spiral-like frequency dependence of their mass, angular momentum and Noether charge as single boson stars. In contrast, sequences of chains with an odd number of constituents show nontrivial loops starting and ending at the flat vacuum. As a consequence, such solutions cannot be uniquely parametrized by a single parameter. We conjecture that all rotating chains correspond to excitations of single boson stars or pairs of them. We also analyze their flat space limit and find that they reduce to chains of $Q$-balls.