The authors present a new method for the inversion of the exponential Radon transform based on the harmonic analysis of the Euclidean motion group of the plane. For a uniform attenuation coefficient \(\mu \in \mathbb{C}\), the exponential Radon transform of a compactly supported real valued function \(f\) over \(\mathbb{R}^{2}\) is defined by \[ \mathcal{T}_{\mu}f(\theta,t)=\int _{\mathbb{R}^{2}}f(\mathbf{x})\delta(\mathbf{x}\cdot \pmb{\theta}-t)e^{\mu \mathbf{x}\cdot\pmb{\theta} ^{\perp}}d\mathbf{x} \] where \(t\in \mathbb{R}\), \(\pmb{\theta}=(\cos \vartheta, \sin \vartheta)\), and \(\pmb{\theta} ^{\perp}=(- \sin \vartheta, \cos \vartheta)\), \(0\leq \vartheta<2\pi\). By multiplying \( \mathcal{T}_{\mu}f(\theta,-r_{1})\) with \(e^{\mu r_{2}}\), \(r_{2}\in \mathbb{R}\), the resulting integral can be expressed as a convolution on the Euclidean motion group. The Euclidean motion group Fourier transform is then used to block diagonalize this modified exponential Radon transform. Further analysis of such a representation leads to new reconstruction algorithms which are numerically implemented and their performance is compared with the filtered backprojection algorithm.