two possibilities:

1. assuming the rebuild was good, check the coolant level.  look in the
radiator, not the expansion bottle.  if you don't have the correct
coolant level, the temp sensor is not getting the right signal and it
tells the motor to go into cold start mode.  that injects excess gas,
dilutes the oil film on the cylinder walls & suddenly, it's consuming
oil.  also check/replace the thermostat & the oxygen sensor.  but i
doubt this would be as bad as a quart a week.

2. you mention replacing rings, but not pistons.  when you replace
rings, you need to re-hone the cylinder wall to break up the glaze layer
for the new rings to bite.  re-honing usually means new pistons.  if you
didn't hone, the rings will never seal and it'll drink oil.  in
addition, all traces of abrasive need to be removed from the cylinder
wall after honing too.  re-built engines almost always have a much
accelerated wear rate compared to factory because this vital step of
cylinder honing abrasive removal is either overlooked or done badly.
excess wear = excess oil consumption.