In a large pot, bring water to a boil. Add green beans and cook until tender yet still crisp, about four minutes. Using a slotted spoon, transfer the beans to a colander. Rinse under cold water and drain. Return the pot of water to a boil and add the potato slices. Cook until tender yet firm, about four minutes. Drain, rinse under cold water, and drain well.
Meanwhile, bring another pot of water to a boil. Add a generous amount of salt to the water and cook linguine al dente, about seven-eight minutes. Add the beans and potatoes during the last minute of cooking the pasta. Drain, saving one cup of the pasta water.
Transfer pasta, green beans and potatoes to a large serving bowl. Add the pesto and just enough of the reserved pasta water to make a fluid sauce. Toss gently and serve.