  We use the cluster variation method (CVM) to investigate the phase structure
of the 3d gonihedric Ising actions defined by Savvidy and Wegner. The
geometrical spin cluster boundaries in these systems serve as models for the
string worldsheets of the gonihedric string embedded in ${\bf Z}^3$. The models
are interesting from the statistical mechanical point of view because they have
a vanishing bare surface tension. As a result the action depends only on the
angles of the discrete surface and not on the area, which is the antithesis of
the standard 3d Ising model.
  The results obtained with the CVM are in good agreement with Monte Carlo
simulations for the critical temperatures and the order of the transition as
the self-avoidance coupling $\kappa$ is varied. The value of the magnetization
critical exponent $\beta = 0.062 \pm 0.003$, calculated with the cluster
variation--Pad\`e approximant method, is also close to the simulation results.
