  We present exact calculations of the Potts model partition function Z(G,q,v)
for arbitrary q and temperature-like variable v on n-vertex strip graphs G of
the triangular lattice for a variety of transverse widths equal to L vertices
and for arbitrarily great length equal to m vertices, with free longitudinal
boundary conditions and free and periodic transverse boundary conditions. These
have the form Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}}
c_{Z,G,j}(\lambda_{Z,G,j})^{m-1}. We give general formulas for N_{Z,G,j} and
its specialization to v=-1 for arbitrary L. The free energy is calculated
exactly for the infinite-length limit of the graphs, and the thermodynamics is
discussed. It is shown how the internal energy calculated for the case of
cylindrical boundary conditions is connected with critical quantities for the
Potts model on the infinite triangular lattice. Considering the full
generalization to arbitrary complex q and v, we determine the singular locus
{\cal B}, arising as the accumulation set of partition function zeros as
m\to\infty, in the q plane for fixed v and in the v plane for fixed q.
