Numerous important integrals involve products of the Fox H-functions and their particular cases. The most comprehensive and universal method for calculating integrals of a single Fox H-function or the product of two Fox H-functions has been developed by O.I. Marichev and has been implemented in Wolfram Mathematica computer algebra system. However, as of now, no general method for finding integrals of the product of three or more Fox H-functions has been implemented in any computer algebra system. This method, based on calculation of multidimensional residues, is the subject of the present article.