I haven't tried bleach yet. I'm in a similar predicament myself. I haven't used anything very long lasting yet, but the foaming drain opener works well. Caustic soda works too, but it is rather harsh.
They make a drain cleaner that has enzymes which digest the organics in the drain (yummy). These are for drains clogged by grease and hair (as opposed to mineral deposits). They aren't harsh and seem to help. Bleach would kill the enzymes. I'm not sure if bleach would help clear a drain. Laundry bleach isn't even that strong for a drain cleaner. You might have sterile clogged drains. :-)
I'm thinking of designing a pressure washer attachment for drain cleaning. I can replace the drain plumbing inside... but I wonder about what's in the pipe on the other side of the foundation.