High-resolution 3D microscopy is a fast advancing field and requires new techniques in image analysis to handle these new datasets. In this work, we focus on detailed 3D segmentation of Dictyostelium cells undergoing macropinocytosis captured on an iSPIM microscope. We propose a novel random walker-based method with a curvature-based enhancement term, with the aim of capturing fine protrusions, such as filopodia and deep invaginations, such as macropinocytotic cups, on the cell surface. We tested our method on both real and synthetic 3D image volumes, demonstrating that the inclusion of the curvature enhancement term can improve the segmentation of the aforementioned features. We show that our method performs better than other state of the art segmentation methods in real microscopy data, and performs better or similar to these methods in synthetic data. We also present an automated seeding method for microscopy data, which, combined with the curvature-enhanced random walker method, enables the rapid segmentation of large time series with minimal input from the experimenter.