A micropump is the heart of any microfluidic device that finds applications in several lab-on-chip devices. Passive micropumps are highly desirable for this purpose due to their ease of integration, low energy requirements, and simplistic design and operation. The design of a plant leaf serves as natural inspiration for the development of an evaporation-assisted passive micropump. The presence of a branching-channel-like venation pattern ensures water distribution to the spongy mesophyll cells, increasing the surface area for evaporation. However, because of its multiscale design and the complexity of the venation pattern, emulating a leaf's design is challenging. Apart from the lack of understanding of design parameters that affect fluid flow, manufacturing limitations impede the development of such bioinspired micropumps. Inspired by the multiscale design of the leaf, in this work we propose a passive micropump mimicking the structure of a leaf. Using evaporation and capillary pressure as the pumping mechanism, our leaf-mimicking micropump consists of a microporous membrane integrated with a branched, fractal channel network resembling a leaf's venation pattern. Our proposed fabrication method is simple, scalable, and inexpensive and uses readily available materials. We demonstrate a significant increase in the fluid flow rate due to the addition of this branched-channel network. We support our experimental observations using an analytical model, wherein we discuss the design parameters that affect the pumping rate. Correspondingly, the performance of these micropumps can be optimized on the basis of intrinsic and extrinsic factors as per the desired applications.