In this paper a general and simple framework is proposed for pricing Asian options under a variety of pricing models, including the Merton model and the Carr- Geman-Madan-Yor model. The approach uses the exponential time integration scheme in combination with a dimensional splitting technique. For discretely observed Asian options, a simple splitting technique is applied, while a Strang splitting strategy is employed to improve the temporal accuracy for continuously observed averages. For European Asian options it is shown that spectral accuracy can be achieved using a Chebyshev discretization. Barrier features are easily enforced and an operator-splitting technique is used to price Amerasians. For the Black-Scholes model we show that employing best rational approximation via Carathéodory-Fejér approximation leads to algorithms with linear complexity in each spatial dimension.