Brief answer: yes, there are a lot of articles where the computations are done. I even don't know where to start. The work of Helena Verrill and Chuck Doran as well as Zagier's lectures in the 1-2-3 of modular forms (mentioned by Scott Carnahan) are a definite must. In my joint work with G. Almkvist and D. van Straten this question is addressed as well and you will find some relevant references there (for example, to Don's Apery-like article). A lot of modular cases with the corresponding Picard--Fuchs differential equations are discussed in my other joint paper with Heng Huat Chan [Mathematika 56:1 (2010) 107--117] and in a recent preprint New analogues of Clausen's identities arising from the theory of modular forms (which I am not allowed to distribute publicly). You may also check Robert S. Maier's arXiv:math/0501425 and his other publications.
I have to add the books which cover in full the hypergeometric differential equations uniformized by modular functions: [J.M. Borwein and P.B. Borwein, Pi and the AGM; A study in analytic number theory and computational complexity (Wiley, New York, 1987)] and [M. Yoshida, Hypergeometric functions, my love. Modular interpretations of configuration spaces, Aspects of Mathematics E32 (Friedr. Vieweg & Sohn, Braunschweig, 1997)].