As genneth said in a comment, any metal around (or maybe somewhat above) room temperature should have a higher heat capacity than $3k_B$ per atom.
Each vibrational degree of freedom (a.k.a. phonon mode) has a heat capacity of $k_B$ as long as the temperature $T$ and vibrational frequency $\nu$ satisfy $k_BT\gg h\nu$ (if this is not satisfied, the heat capacity is less than $k_B$). There are three phonon modes per atom, so phonons give you $3k_B$, as long as the temperature is high enough. For example, in gold, all the phonon frequencies are less than 5 THz; 5 THZ corresponds to 240K; so at room temperature the phonon heat capacity is almost $3k_B$ (but a bit less).
(I chose gold as an example because its atoms are heavy so they vibrate slowly. Metals with lighter atoms have higher vibration frequencies so a higher required temperature to get the full $3k_B$.)
On top of the phonons, a metal also has heat capacity from kinetic energy of the free electrons. So altogether it can be more than $3k_B$.
For example, I looked up gold's heat capacity (.128 or .129 J/gK) and atomic mass (196.97) and got $3.03k_B$ to $3.06k_B$ per atom.
(I'm a bit surprised it's not higher, since each atom should contribute at least one free electron, and a free electron would be expected to have $1.5k_BT$ of translational kinetic energy. I guess it's too simplistic to treat the electrons like non-interacting free particles. For example, maybe there is a ceiling on electron kinetic energy because of the band structure, or because of velocity-dependent phonon scattering? I'm not sure.)
Other possible degrees of freedom that provide extra heat capacity in some solids include plasmons, magnons, excitons, polaronic excitations, and many others. :-)