Given a subset $S$ of the non-identity elements of the dihedral group of order $2m$, is it possible to order the elements of $S$ so that the partial products are distinct? This is equivalent to the sequenceability of the group when $|S| = 2m-1$ and so it is known that the answer is yes in this case if and only if $m>4$. We show that the answer is yes when $|S| \leq 9$ and $m$ is an odd prime other than 3, when $|S| = 2m-2$ and $m$ is even or prime, and when $|S| = 2m-2$ for many instances of the problem when $m$ is odd and composite. We also consider the problem in the more general setting of arbitrary non-abelian groups and discuss connections between this work and the concept of strong sequenceability.