The densities do increase as you move down the table within a group, but the density doesn't uniformly increase or decrease as you move across within a period (see the Wolfram graphic linked below for a 3D plot - select the density as a property). The densities increase as you move down a group for the obvious reason: the atomic masses increase (and the average atomic radii don't increase rapidly such that the average atomic volumes increase as fast as the masses).
The density trends on moving left to right across within a period are more complicated. Of course, the masses are increasing, but much more slowly than when you move down within a group. However, the strength of bonding (which goes inversely with the average atomic volumes) varies more rapidly and has a couple of maxima within the full periods. Consider, for example, the third-row transition metals (period 6) or first-row p-block elements (period 2). The bonding strength maximizes for the top-row p-block elements at carbon (carbon makes 2 bonds per atom, each C makes four bonds but the bonds are shared between 2 atoms - hence 2 bonds per atom. In the third-row T.M.s the bonding strength peaks at W, and the density peaks at Os (because of the higher mass, slightly smaller radius, and nearly as strong net bonding).
