Heh, heh.. at first I thought MERTON was back...
Yeah, grinding welds is usually not a good idea. In addition to the heat-treatment problems mentioned, the grinding usually leaves rough surface-stratches which are stress-risers and prone to cracking. You'll need to polish up and make the surface smooth as well.
Rather than grinding off the welds, you want to cut instead. Use sharp carbide cutters with a high-torque die-grinder instead of the Dremel. The lower RPM heats up the metal the lowest amount for the material removed. Also have the joint freely bathed & rinsed in cutting-oil as you're cutting as well.