  In 1979, Shearer and Kleitman conjectured that there exist $\lfloor n/2
\rfloor+1$ orthogonal chain decompositions of the hypercube $Q_n$, and
constructed two orthogonal chain decompositions. In this paper, we make the
first non-trivial progress on this conjecture since by constructing three
orthogonal chain decompositions of $Q_n$ for $n$ large enough. To do this, we
introduce the notion of "almost orthogonal symmetric chain decompositions". We
explicitly describe three such decompositions of $Q_5$ and $Q_7$, and describe
conditions which allow us to decompose products of hypercubes into $k$ almost
orthogonal symmetric chain decompositions given such decompositions of the
original hypercubes.
