A common requirement for spatial modeling is the development of an appropriate correlation structure. Although the assumption of isotropy is often made for this structure, it is not always appropriate. A conventional practice when checking for isotropy is to informally assess plots of direction-specific sample (semi)variograms. Although a useful diagnostic, these graphical techniques are difficult to assess and open to interpretation. Formal alternatives to graphical diagnostics are valuable, but have been applied to a limited class of models. In this article we propose a formal approach to test for isotropy that is both objective and valid for a wide class of models. This approach, which is based on the asymptotic joint normality of the sample variogram, can be used to compare sample variograms in multiple directions. An L₂-consistent subsampling estimator for the asymptotic covariance matrix of the sample variogram is derived and used to construct a test statistic. A subsampling approach and a limiting chi-squared approach are developed to obtain p values of the test. Our testing approach is purely nonparametric in that no explicit knowledge of the marginal or joint distribution of the process is needed. In addition, the shape of the random field can be quite irregular. The results apply to regularly spaced data as well as to irregularly spaced data when the point locations are generated by a homogeneous Poisson process. A data example and simulation experiments demonstrate the efficacy of the approach.