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The growth rate of the large-scale structure of the Universe has been advocated as the observable par excellence for testing gravity on cosmological scales. By considering linear-order deviations from general relativity, we show that corrections to the growth rate, f, can be expressed as an integral over a "source" term, weighted by a theory-independent "response kernel." This leads to an efficient and accurate "plug-and-play" expression for generating growth rates in alternative gravity theories, bypassing lengthy theory-specific computations. We use this approach to explicitly show that f is sensitive to a degenerate combination of modified expansion and modified clustering effects. Hence the growth rate, when used in isolation, is not a straightforward diagnostic of modified gravity. [ABSTRACT FROM AUTHOR]