  We examine a specific category of eigenfunctions of the lattice Laplacian on
$\{p,q\}$-tessellations of the Poincar\'e disk that bear resemblance to plane
waves in the continuum case. Our investigation reveals that the lattice
eigenmodes deviate from the continuum solutions by a factor that depends solely
on the local inclination of the vertex in relation to the wave's propagation
direction. This allows us to compute certain eigenfunctions by numerical and
analytical methods. For various special cases we find explicit exact
eigenfunctions and their eigenvalues on the infinite lattice.
