The mixed alkali clusters Li$_n$Na$_{8-n}$, Li$_n$K$_{8-n}$, and Na$_n$K$_{8-n}$ were studied by Kohn-Sham theory with a gradient corrected (PBE) exchange-correlation functional. Geometry optimization was done with a Tabu Search algorithm to find possible global minima, followed by local optimization with a quasi-Newton method. For the pure clusters and all Na$_n$K$_m$ mixed clusters, the predicted global minimum is a dodecahedron having D$_{2d}$ symmetry in its ideal form. There are various structures among the predicted global minima of Li$_n$Na$_m$ and Li$_n$K$_m$: they all have in common that the Li$_n$ subunit achieves maximal coordination and forms a compact core at, or near, the center of the cluster. The clusters Li$_n$Na$_m$ and Li$_n$K$_m$ all have near zero asphericity, in line with the prediction of the ellipsoidal jellium model for 8 electrons, and the asphericity in nonzero but small in all other clusters. The clusters Li$_4$Na$_4$, Na$_4$K$_4$, and K$_4$Na$_4$ each have the four atoms of the lighter element near the center of mass and these clusters are more stable than those of other compositions. Calculated ionization potentials and static dipole polarizabilities agree rather well with experimental values but do not allow a structure assignment to be made.