We numerically investigate the influence of roughness in layer thicknesses on the properties of hyperbolic metamaterials (HMMs). We show that random spatial variation of dielectric and metal layer thicknesses, similar to what occurs during actual structure fabrication, leads to dramatic absorption increase compared to an ideal, smooth-layer HMM; the absorption increases more strongly when roughness is induced throughout the HMM rather than in its surface layer only. Hence, we have found that moderate surface roughness does not deteriorate the HMM functionality, at least in absorption-related applications, thus eliminating the challenge of ultrasmooth metal layer fabrication. More severe roughness can also prove useful in the design of inexpensive HMM-based broadband absorbers.