ABSTRACT Next-generation radio surveys are expected to detect tens of millions of active galactic nuclei (AGNs) with a median redshift of $z$ ≥ 1. Beyond targeted surveys, the vast majority of these objects will not have spectroscopic redshifts, while photometric redshifts for high-redshift AGNs are of limited quality, and even then require optical and infrared photometry. We propose a new approach to measure the redshifts of lobed radio galaxies based exclusively on radio-frequency imaging and broad-band radio photometry. Specifically, our algorithm uses the lobe flux density, angular size and width, and spectral shape to derive probability density functions for the most likely source redshift based on the Radio AGN in Semi-analytic Environments dynamical model. The full physically based model explains 70 per cent of the variation in the spectroscopic redshifts of a high-redshift (2 < $z$ < 4) sample of radio AGNs, compared to at most 27 per cent for any one of the observed attributes in isolation. We find that upper bounds on the angular size, as expected for unresolved sources, are sufficient to yield accurate redshift measurements at $z$ ≥ 2. The error in the model upon calibration using at least nine sources with known spectroscopic redshifts is <14 per cent in redshift (as 1 + $z$) across all redshifts. We provide the python code for the calculation and calibration of our radio continuum redshifts in an online library.