There are several recent results on non-trivial bounded functions $f$ that *are* orthogonal, i.e. 
$$\sum\_{n\leq x} \mu(n)f(n)=o(x),$$
to the Mobius function $\mu(n)$. These include the results of [Green and Tao on Nilsequences](http://www.arxiv.org/abs/0807.1736) (which includes information on the history of the problem) and [Green on bounded depth circuits](http://www.https:arxiv.org/abs/1103.4991). The lecture notes by [Peter Sarnak on Mobius randomness](https://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf) is an interesting overview of the subject. 


Sarnak mentions there that it is easy to construct counter examples, or examples which decay very slowly. Of course, its easy to do so if $f$ is sparsely supported or has partial sums that are the inverse Mellin transform of a function having positive powers of $\zeta(s)$ as factors. 



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> What are some examples of bounded, non-vanishing (or at least positive density) functions that *are not* orthogonal to $\mu(n)$, and which don't come from $\zeta(s)$ in an obvious way? 
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Interesting examples of functions that are not orthogonal to the Mobius function?
This question has been considered by Lemanczyk and others, and Lemanczyk developed a quite general way to produce dynamical systems which are not disjoint from Mobius (unfortunately or fortunately, depending on ones preferences, they are of positive entropy).
The basic idea is to use the square free flow constructed by Sarnak (which Terry have mentioned) and to find some symbolic system which "approximates" this flow for large times.
See for example the detailed survey - <https://arxiv.org/abs/1410.1673>
In the end, the fact that there are systems which are not Mobius disjoint is not that surprising - Sarnak has computed the entropy of the Mobius flow and show it positive, general ergodic theory (i.e. the construction of the Bernoulli factor by Sinai/Ornstein) would tell you it is "easy" to correlate with Mobius.


Just another related interesting comment - Bourgain has constructed a system with positive entropy which is Mobius disjoint.


