Fuzzy cognitive maps (FCMs) have demonstrated considerable success in time series forecasting and are adept at handling uncertainties and capturing the dynamics of complex systems. Nevertheless, challenges still remain in the handling of multivariate high-dimensional time series using a time-effective learning algorithm. This article introduces multiple-input multiple-output randomized high-order FCM (MRHFCM), a new methodology for predicting high-dimensional time series in multiple-input-multiple-output systems. MRHFCM represents a hybrid method that combines data embedding transformation, randomized high-order FCM (R-HFCM), and an echo state network. The core of MRHFCM involves a cascade of R-HFCMs termed the CR-HFCM model. Each CR-HFCM comprises three layers: 1) the input layer, 2) reservoir (internal layer), and 3) output layer. Notably, only the output layer is trainable, employing the least squares minimization algorithm. The weights within each subreservoir are randomly chosen and remain unchanged throughout the training procedure. Three real-world high-dimensional datasets are utilized to assess the performance of the proposed MRHFCM method. The results obtained reveal that our approach outperforms some existing baseline and state-of-the-art machine learning and deep learning forecasting techniques in terms of both accuracy and parsimony.