The motivation for investigating Stokes flow through an array of staggered circular cylinders in a topographically patterned microchannel arises from a wide range of applications, including microfluidic devices, especially for cooling purposes, enhancing mass transport, noise control during aircraft landing, heat transfer enhancement, and so forth. Two-dimensional viscous incompressible, pressure-driven, creeping flow at low Reynolds number around a series of staggered circular cylinders in a topographically patterned microchannel is investigated numerically by using the boundary element method (BEM) based on a non-primitive variables approach. The non-primitive variables approach refers to the combination of stream function and vorticity variables. The Stokes equation is used to govern the flow of viscous fluid through a topographically patterned microchannel. We restrict ourselves to rectangular stepped asymmetric geometries. Moreover, we have assumed that a number of cylinders of equal diameter are present at the staggered position in the path of flow. We studied streamline and vorticity contour plots, velocity profiles, pressure gradients, and the shear stresses with varied step height, step width, step frequency, and the radius of the cylinder, to get a complete comprehension of flow dynamics. Additionally, the proposed investigation holds several potential applications, such as drug capsule delivery systems, hemodynamics, bio-MEMS technology, and so forth.