We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter's theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.