At this moment, there are no widely known explanations that are as "ready to be used" as the anthropic would-be explanation.
There are of course vague ideas how the problem could be solved but no full framework how to incorporate particle physics and predict a tiny C.C. has been proposed. The partial proposals include:
The supersymmetric cancellations
Low-scale supersymmetry reduces the natural prediction of the C.C. from $O(1)$ to $O(10^{-60})$ in Planck units which seems progress but it's still very far from the required $O(10^{-123})$. Despite the fact that SUSY seems to "solve" one half of the problem on the log scale, in some sense, it makes the problem qualitatively worse because one can be "more certain" than the $O(10^{-60})$ piece which is still way too large survives.
No-scale supergravity is a framework to guarantee that the leading term to the cosmological constant cancels. In general, spontaneously broken supersymmetry hasn't been able to explain the smallness of the C.C.
Mirror matter
There are various highly speculative articles suggesting that there exists a "mirror world" whose zero-point energies cancel against those in our world. Such worlds are probably unstable and suffer from many other problems.
Link to the neutrino masses
The C.C. expressed as a vacuum energy density is comparable to the fourth power of the neutrino masses and the neutrinos are the lightest massive particles we know. It could be that there is something wrong about the ways how we apply - or fail to apply - the Renormalization Group on the running of the C.C. It may be that there is some mechanism that makes a theory inconsistent if the C.C. at the neutrino scale - which is also the C.C. at zero energies - is much larger than the neutrino scale.
Seesaw mechanisms
It just happens that the C.C. is approximately $m_{TeV}^8 / m_{Planck}^4$, which is linked to the formula for the neutrino masses. So on the log scale, the C.C. is on the opposite side from the natural TeV-scale energy density (e.g. predicted by broken SUSY) than the Planck density on the other side.
There exist highly speculative proposals how to derive this seesaw formula from a deeper mechanism, for example, from some mixing between different vacua, tunneling, and so on.
Others and appraisal
There also exist various papers on self-adjusting cosmological constants that don't work, as far as I know. If one looks at what is known, it's fair to say that the anthropic explanation of the C.C. is the only available game in town. However, in this particular case, it's totally plausible that a much better explanation will be found in the future. In this case, there's no real evidence that all the alternative explanations have been mapped or ruled out.
