The interpretation of molecular translational diffusion as measured by pulsed gradient spinecho NMR (PGSE NMR) can be complicated by the presence of chemical exchange and/or dipolar cross-relaxation (including relayed cross-relaxation via spin diffusion). The magnitude of influence depends on the kinetics of exchange and/or dipolar cross-relaxation present within the system as well as the PGSE NMR sequences chosen for the measurements. Firstly, we present an exchange induced zero-crossing phenomenon for signal attenuation of water in lipidic cubic phases (LCPs, formed by a mixture of monoolein and water) in the presence of pulsed gradients observed using a standard STimulated Echo (STE) sequence. This magnetization exchange induced zero-crossing phenomenon, a pseudo negative diffractionlike feature, resembles that reported previously for restricted diffusion when locally anisotropic pores are polydisperse or randomly oriented. We then demonstrate the elimination of these exchange and/or dipolar cross-relaxation induced effects with the use of a chemical shift selective STE (CHESS-STE) sequence, adapted from the previously reported band-selective short transient STE (BEST-STE) sequence, along with results obtained from the bipolar pulse pair STE (BPP-STE) sequence for comparison. The CHESS-STE sequence introduced here represents a generic form of PGSE NMR sequences for obtaining water diffusion coefficients free from influence of exchange and/or dipolar cross-relaxation in complex systems. It has potential applications in measuring translational diffusion of water in biopolymer mixtures as well as probing microscopic structure in materials via water restricted diffusion measured by PGSE NMR, particularly when the potential presence of exchange/cross-relaxation is of concern.