A new variety of the "soliton management" in heterogeneous optical media is proposed. The system is composed as a periodic chain of nonlinear fibers with negligible intrinsic group-velocity dispersion (GVD), alternating with sections of unchirped fiber Bragg gratings (FBGs) operating in the reflection regime. Losses due to incomplete reflection are compensated by linear amplifiers. The model may apply to fiber-optic telecommunication links with periodically installed FBG modules, and it may be used for the design of laser setups. By means of extended simulations, we identify small regions in the underlying parameter space where this model, featuring the periodic separation of the Kerr nonlinearity and FBG-induced GVD (hence the name of the "split-step" system), supports stable transmission of RZ (return-to-zero) pulses, i.e., quasi-solitons. The effect of nonzero fiber's GVD on the stable transmission regime is considered too. Moderately unstable (partly usable) transmission regimes are found in larger regions of the parameter space; they may be of two different types, with the average nonlinearity either undercompensating or overcompensating the GVD. Interactions between the stable RZ pulses are also studied, leading to the identification of a minimum separation between them necessary for the suppression of interaction effects.