Increasing attention is devoted to the use of nanomechanics as a marker of various pathologies. Atomic force microscopy (AFM) is one of the techniques that could be applied to quantify the nanomechanical properties of living cells with a high spatial resolution. Thus, AFM offers the possibility to trace changes in the reorganization of the cytoskeleton in living cells. Impairments in the structure, organization, and functioning of two main cytoskeletal components, namely, actin filaments and microtubules, cause severe effects, leading to cell death. That is why these cytoskeletal components are targets for antitumor therapy. This review intends to describe the gathered knowledge on the capability of AFM to trace the alterations in the nanomechanical properties of living cells induced by the action of antitumor drugs that could translate into their effectiveness.