Bounded, non-constant harmonic functions: how far are they from existing?

If an $L^2$ treatment is feasible one could try Fourier transform. On a purely formal level any $f:\ {\mathbb Z}^2\to{\mathbb C}$ has a Fourier transform $\hat f:\ T^2\to {\mathbb C}$ which is nothing else but the doubly periodic function $$f(x,y):=\sum_{k,l} f(k,l)e^{i(k x+ly)}\qquad\bigl((x,y)\in ({\mathbb R}/(2\pi))^2\ \bigr)\ .$$ One easily checks that $$\widehat{Tf}(x,y)={1\over2}(\cos x+\cos y)\hat f(x,y)\ ,$$ so that in the "Fourier domain" the "convolution operator" $T$ acts as multiplication with a simple function. As $f$ was assumed real its Fourier transform has certain symmetry properties which might simplify the resulting expressions.