A method to calculate the bound states of three atoms without resorting to an explicit partial wave decomposition is presented. The differential form of the Faddeev equations in the total angular momentum representation is used for this purpose. The method utilizes Cartesian coordinates combined with the tensor trick preconditioning for large linear systems and Arnoldi's algorithm for eigenanalysis. As an example, we consider the He₃ system in which the interatomic force has a very strong repulsive core requiring the inclusion of a large number of partial waves to achieve convergence. To improve convergence and stability in the Arnoldi's algorithm, we modify the system of equations by introducing a background potential that removes the large number of unphysical eigenvalues stemming from the hard core. The introduction of such a modifying potential does not affect the physical spectrum of the system. The results obtained by solving the modified, three-dimensional, Faddeev equations compare favorably with other results in the field.