Accurate estimates of genome-wide rates and fitness effects of new mutations are essential for an improved understanding of molecular evolutionary processes. Although eukaryotic genomes generally contain a large noncoding fraction, functional noncoding regions and fitness effects of mutations in such regions are still incompletely characterized. A promising approach to characterize functional noncoding regions relies on identifying accessible chromatin regions (ACRs) tightly associated with regulatory DNA. Here, we applied this approach to identify and estimate selection on ACRs in Capsella grandiflora, a crucifer species ideal for population genomic quantification of selection due to its favorable population demography. We describe a population-wide ACR distribution based on ATAC-seq data for leaf samples of 16 individuals from a natural population. We use population genomic methods to estimate fitness effects and proportions of positively selected fixations (α) in ACRs and find that intergenic ACRs harbor a considerable fraction of weakly deleterious new mutations, as well as a significantly higher proportion of strongly deleterious mutations than comparable inaccessible intergenic regions. ACRs are enriched for expression quantitative trait loci (eQTL) and depleted of transposable element insertions, as expected if intergenic ACRs are under selection because they harbor regulatory regions. By integrating empirical identification of intergenic ACRs with analyses of eQTL and population genomic analyses of selection, we demonstrate that intergenic regulatory regions are an important source of nearly neutral mutations. These results improve our understanding of selection on noncoding regions and the role of nearly neutral mutations for evolutionary processes in outcrossing Brassicaceae species.