Computer-generated holography (CGH) holds transformative potential for a wide range of applications, including direct-view, virtual and augmented reality, and automotive display systems. While research on holographic displays has recently made impressive progress, image quality and eye safety of holographic displays are fundamentally limited by the speckle introduced by coherent light sources. Here, we develop an approach to CGH using partially coherent sources. For this purpose, we devise a wave propagation model for partially coherent light that is demonstrated in conjunction with a camera-in-the-loop calibration strategy. We evaluate this algorithm using light-emitting diodes (LEDs) and superluminescent LEDs (SLEDs) and demonstrate improved speckle characteristics of the resulting holograms compared with coherent lasers. SLEDs in particular are demonstrated to be promising light sources for holographic display applications, because of their potential to generate sharp and high-contrast two-dimensional (2D) and 3D images that are bright, eye safe, and almost free of speckle.