Copulas and other multivariate models can give joint exceedance probabilities for multivariate events in the natural environment. However, the choice of the most appropriate multivariate model may not always be evident in the absence of knowledge of dependence structures. A simple nonparametric alternative is to approximate multivariate dependencies using “line mesh distributions”, introduced here as a data-based finite mixture of univariate distributions defined on a mesh of L  = C( m , 2) lines extending through Euclidean n -space. That is, m data points in n -space define a total of L lines, where C() denotes the binomial coefficient. The utilitarian simplicity of the method has attraction for joint exceedance probabilities because just the data and a single bandwidth parameter within the 0, 1 interval are needed to define a line mesh distribution. All bivariate planes in these distributions have the same Pearson correlation coefficients as the corresponding data. Marginal means and variances are similarly preserved. Using an example from the literature, a 5-parameter bivariate Gumbel model is replaced with a 1-parameter line mesh distribution. A second illustration for three dimensions applies line mesh distributions to data simulated from a trivariate copula.