For advanced turbomachinery development, there is increasing interest to carry out unsteady analyses for flows through multiple bladerows during a design stage. Even with the huge increase in computer processing power currently available, direct unsteady calculations in a whole annulus domain are still very time consuming. Efficient alternative methods with truncations in time and/or in space have been developed for unsteady turbomachinery flows in the past 20 years or so, but they all are associated with related limitations. The present development is motivated to maintain as many modeling fidelities of direct unsteady solution methods as possible while still have a significant speed-up. To this end, a new steady-solution-like unsteady time-domain methodology has been developed for bladerow interactions. No circumferential domain truncation is required so that a direct periodic (repeating) condition can be applied. For each mesh cell, the temporal gradient term as required to balance the discretized unsteady flow equation is obtained by specially sequenced spatial variations of corresponding cells in multiple-passages. Consequently, the rotor–stator interface treatment becomes completely compatible to that of a direct unsteady solution. Thus a fully conservative interface is easily achieved, in contrast to existing truncated models where interface treatments tend to be complicated and nonconservative. The simultaneous solution procedure with the space–time gradient (STG) link enables an unsteady flow solution to converge at a rate compatible to a steady solution. The background, motivation/justification, basic methodology, and some preliminary verifications are described in this paper as Part I. Further validations and applications with extension to more complex configurations and flow conditions will be presented in Part II.