Astronomy depends on ever-increasing computing power. Processor clock rates have plateaued, and increased performance is now appearing in the form of additional processor cores on a single chip. This poses significant challenges to the astronomy software community. Graphics processing units (GPUs), now capable of general-purpose computation, exemplify both the difficult learning curve and the significant speedups exhibited by massively parallel hardware architectures. We present a generalized approach to tackling this paradigm shift, based on the analysis of algorithms. We describe a small collection of foundation algorithms relevant to astronomy and explain how they may be used to ease the transition to massively parallel computing architectures. We demonstrate the effectiveness of our approach by applying it to four well-known astronomy problems: Hoegbom clean, inverse ray-shooting for gravitational lensing, pulsar dedispersion and volume rendering. Algorithms with well-defined memory access patterns and high arithmetic intensity stand to receive the greatest performance boost from massively parallel architectures, while those that involve a significant amount of decision-making may struggle to take advantage of the available processing power.