We report on growth of 3C-SiC by sublimation process in vacuum with the aim to ultimately select conditions for single polytype growth of bulk crystals. The 3C polytype occurrence, growth mechanism and structure evolution have been in the focus of the study. To gain understanding of the initial formation of the cubic polytype, growth was performed on various substrates, such as 6H- and 4H-SiC (on-axis and vicinal), as well as freestanding 3C-SiC wafers. The growth configuration used allowed a high growth rate, e.g. up to 200 (m/h, respectively very thick layers. The grown material was studied by means of optical microscopy, AFM and HRTEM. 6H-SiC (0001) Si-face substrates may be a good choice if the 3C nucleation is well controlled, which can be achieved by selecting the initial temperature ramp up and substrate orientation. These growth conditions limit the number of nucleation centers and decrease the defective boundaries.