  We study the non-equilibrium dynamics of kicked Ising models in $1+1$
dimensions which have interactions alternating between odd and even bonds in
time. These models give rise to time-evolution equivalent to quantum circuits
having both the global property of tri-unitarity (three 'arrows of time') and
also the local property of second-level dual-unitarity, which constrains the
behavior of pairs of local gates underlying the circuit under a space-time
rotation. We identify a broad class of initial product states wherein the
effect of the environment on a small subsystem can be exactly represented by
influence matrices with simple Markovian structures, resulting in the
subsystem's full dynamics being efficiently computable. We further find
additional conditions under which the dynamics of entanglement can be solved
for all times, yielding rich phenomenology ranging from linear growth at half
the maximal speed allowed by locality, followed by saturation to maximum
entropy (i.e., thermalization to infinite temperature); to entanglement growth
with saturation to extensive but sub-maximal entropy. Our findings extend our
knowledge of interacting quantum systems whose thermalizing dynamics can be
efficiently and analytically computed, going beyond the well-known examples of
integrable models, Clifford circuits, and dual-unitary circuits.
