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Differential difference algebras are generalizations of polynomial algebras,quantum planes, and Ore extensions of automorphism type and of derivation type.In this paper, we investigate the Gelfand-Kirillov dimension of a finitelygenerated module over a differential difference algebra through a computationalmethod: Gr\"obner-Shirshov basis method. We develop the Gr\"obner-Shirshovbasis theory of differential difference algebras, and of finitely generatedmodules over differential difference algebras, respectively. Then, viaGr\"obner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillovdimensions of cyclic modules and finitely generated modules over differentialdifference algebras.

Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras