The structural information in high-dimensional transposable data allows us to write the data recorded for each subject in a matrix such that both the rows and the columns correspond to variables of interest. One important problem is to test the null hypothesis that the mean matrix has a particular structure without ignoring the dependence structure among and/or between the row and column variables. To address this, we develop a generic and computationally inexpensive nonparametric testing procedure to assess the hypothesis that, in each predefined subset of columns (rows), the column (row) mean vector remains constant. In simulation studies, the proposed testing procedure seems to have good performance and, unlike simple practical approaches, it preserves the nominal size and remains powerful even if the row and/or column variables are not independent. Finally, we illustrate the use of the proposed methodology via two empirical examples from gene expression microarrays.