Disease risk variants identified by GWAS are predominantly noncoding, suggesting that gene regulation plays an important role. eQTL studies in unaffected individuals are often used to link disease-associated variants with the genes they regulate, relying on the hypothesis that noncoding regulatory effects are mediated by steady-state expression levels. To test this hypothesis, we developed a method to estimate the proportion of disease heritability mediated by the cis-genetic component of assayed gene expression levels. The method, gene expression co-score regression (GECS regression), relies on the idea that, for a gene whose expression level affects a phenotype, SNPs with similar effects on the expression of that gene will have similar phenotypic effects. In order to distinguish directional effects mediated by gene expression from non-directional pleiotropic or tagging effects, GECS regression operates on pairs of cis SNPs in linkage equilibrium, regressing pairwise products of disease effect sizes on products of cis-eQTL effect sizes. We verified that GECS regression produces robust estimates of mediated effects in simulations. We applied the method to eQTL data in 44 tissues from the GTEx consortium (average NeQTL = 158 samples) in conjunction with GWAS summary statistics for 30 diseases and complex traits (average NGWAS = 88K) with low pairwise genetic correlation, estimating the proportion of SNP-heritability mediated by the cis-genetic component of assayed gene expression in the union of the 44 tissues. The mean estimate was 0.21 (s.e. = 0.01) across 30 traits, with a significantly positive estimate (p < 0.001) for every trait. Thus, assayed gene expression in bulk tissues mediates a statistically significant but modest proportion of disease heritability, motivating the development of additional assays to capture regulatory effects and the use of our method to estimate how much disease heritability they mediate.