Engine/drivetrain vibration?

A cast crank 340 that was not internally balanced by the machine shop needs a properly located small weight on the converter or a B&M flexplate - 10235 for a 727. Jegs and Summit sell them for $70-$80 each.
Those have a section cut out 180* opposite of the weight location on the OEM 340 cast crank converter. (Accomplishes the same thing and probably saves a few cents in mfg. costs.)

If all of the weights were removed from the converter in use, it has become "neutral" balanced.

Double-check to make sure the balancer/damper has a raised ring with a counterweight and "For 340 cast crank only" cast into it. If that's the original crank and balancer combo, it's a cast crank. Refer to the link posted by Convertriple 73.

Did you use the OEM pistons & rods for the rebuild?