This article considers the problem of testing conditional independence between two random vectors \(bm X\) and \(\bm Y\) given a confounding random vector \(\bm Z\). An exchangeable-pairs framework is introduced through which the conditional independence testing problem is reformulated as a two-sample testing problem. The framework is motivated by ideas from the model-X literature and is based on a fundamental exchangeability property that holds under the null hypothesis of conditional independence. An energy-distance/maximum mean discrepancy type measure is employed on the resulting exchangeable pairs to quantify departures from conditional independence. A consistent estimator of the proposed discrepancy measure is constructed and its theoretical properties are established under general assumptions. A conditional independence test is then developed using this estimator as a test statistic and is calibrated through a suitable resampling procedure. It is shown that the proposed test is consistent against fixed alternatives, possesses nontrivial asymptotic power against local contiguous alternatives, attains the minimax separation rate for detecting alternatives characterized by the proposed discrepancy measure, and remains consistent when the data dimension diverges with the sample size. The effect of estimating the conditional distribution used to generate the exchangeable pairs is also investigated, and condition under which validity and power properties are preserved is established. Extensive simulation studies demonstrate that the proposed procedure performs competitively with some state-of-the-art methods.