Cyclic peptides have received increasing attention over the recent years as potential therapeutics for "undruggable" targets. One major obstacle is, however, their often relatively poor bioavailability. Here, we investigate the structure-permeability relationship of 24 cyclic decapeptides that share the same backbone N-methylation pattern but differ in their side chains. The peptides cover a large range of values for passive membrane permeability as well as lipophilicity and solubility. To rationalize the observed differences in permeability, we extracted for each peptide the population of the membrane-permeable conformation in water from extensive explicit-solvent molecular dynamics simulations and used this as a metric for conformational rigidity or "prefolding." The insights from the simulations together with lipophilicity measurements highlight the intricate interplay between polarity/lipophilicity and flexibility/rigidity and the possible compensating effects on permeability. The findings allow us to better understand the structure-permeability relationship of cyclic peptides and extract general guiding principles.