Characteristic function-based goodness-of-fit tests are suggested for multivariate distributions. The test statistics, which are straightforward to compute, are defined as two-sample criteria measuring discrepancy of empirical characteristic functions between multivariate ranks of the original observations and the ranks obtained from an artificial sample generated from the reference distribution under test. Multivariate ranks are constructed using the theory of the optimal measure transport, thus rendering the tests of a simple null hypothesis distribution-free, while bootstrap approximations are still necessary for testing composite null hypotheses. Asymptotic theory is developed and a simulation study, concentrating on comparisons with previously proposed tests of multivariate normality, demonstrates that the method performs well in finite samples. The broad applicability of the proposed methods is illustrated through an application to a real dataset.