A method is described for deriving the sampling moments of means of random vectors obtained by sampling without replacement from a finite $k$-variate population of $n$ vector members. A table of results is presented listing the moments of order less than or equal to six as a function of the population moments. These moments were originally derived, in a less general form, in the course of developing the Simplex-Sum Designs discussed in [1]. Their possible wider applicability to sampling problems, however, motivated the extension of the work to the general formulas given here.