A three-dimensional system of interacting spins typically develops static long-range order when it is cooled. If the spins are quantum (S=1/2), however, novel quantum paramagnetic states may appear. The most highly sought state among them is the resonating-valence-bond state, in which every pair of neighbouring quantum spins forms an entangled spin singlet (valence bonds) and these singlets are quantum mechanically resonating among themselves. Here we provide an experimental indication for such quantum paramagnetic states existing in frustrated antiferromagnets, Zn(x)Cu(4-x)(OD)(6)Cl(2), where the S=1/2 magnetic Cu2+ moments form layers of a two-dimensional kagome lattice. We find that in Cu(4)(OD)(6)Cl(2), where distorted kagome planes are weakly coupled, a dispersionless excitation mode appears in the magnetic excitation spectrum below approximately 20 K, whose characteristics resemble those of quantum spin singlets in a solid state, known as a valence-bond solid, that breaks translational symmetry. Doping with non-magnetic Zn2+ ions reduces the distortion of the kagome lattice, and weakens the interplane coupling but also dilutes the magnetic occupancy of the kagome lattice. The valence-bond-solid state is suppressed, and for ZnCu(3)(OD)(6)Cl(2), where the kagome planes are undistorted and 90% occupied by the Cu2+ ions, the low-energy spin fluctuations become featureless.