  We present a one-parameter family of periodic packings of regular tetrahedra,
with the packing fraction $100/117\approx0.8547$, that are simple in the sense
that they are transitive and their repeating units involve only four
tetrahedra. The construction of the packings was inspired from results of a
numerical search that yielded a similar packing. We present an analytic
construction of the packings and a description of their properties. We also
present a transitive packing with a repeating unit of two tetrahedra and a
packing fraction $\frac{139+40\sqrt{10}}{369}\approx0.7194$.
