Recently, several local ground‐motion prediction equations (GMPEs) for peak ground acceleration (PGA) and peak ground velocity (PGV) have been developed in northern Iran, most of which are for the horizontal component of seismic motion. There are also few relations of the vertical component of seismic motion. However, there is not any relation for prediction of peak ground displacement (PGD) in this region. In this article, 463 three‐component strong‐motion records from 107 events occurred between 1976 and 2016 with and epicentral distances up to 100 km are selected from a list recorded by the Building and Housing Research Center (BHRC) of Iran, the Disaster and Emergency Management Presidency of Turkey (AFAD), and the Consortium of Organizations for Strong‐Motion Observation Systems (COSMOS) data for deriving GMPEs for both horizontal and vertical motions. Several affecting parameters such as the earthquake magnitude, source‐to‐site distance, site class, and faulting mechanism for different components of peak ground motions are considered as the predictor variables of GMPEs. An evolutionary algorithm, namely Prefix Gene Expression Programming (PGEP), which can present GMPEs for PGA, PGV, and PGD without any predetermined regression models, is used. These presented models are evaluated by residuals, errors, and goodness‐of‐fit measures and then compared with the available GMPEs in the literature, which proves the derived GMPEs relying on PGEP models are consistent with the other relations previously developed with the use of the local and regional data. However, the present equations are more complete and thus can measure the site ground‐motion parameters for the predesign goals more reliably.