I heard one thing to be cautious, when drinking a lot of reverse osmosis water, is that it can flush too many minerals out of your body. So you may need to take them in again via supplement.
I second that.
Let me be a wise-ass for a second:

Kidney stones are often a result of too high Magnesium/Cacium/Vitamin-D uptake (only in excessive amounts, though) and/or high Oxalate ingestion.
Citric Acid is actually functioning as a basic agent (alkalic) in the body, thus reducing Oxalic Acid in the Kidneys. It's effective in Kidney stone prevention.
However, Citric Acid is binding to heavy metals and also able to pass through the blood/brain barrier, so better avoid it.
Vitamin C (Ascorbic Acid) is also said to be working like CA, only a little less effective.
(DYOR)
Great conversation going, guys. Keep it going!

One thing I would comment on is that RO water is still going to be much better for you in the long run than tap. You can always eat well and take supplements to boost your minerals.
Case in point, my tap water comes from a natural fed aquifer and is around 85 ppm, which is pretty darn good compared to our country's standard average of 200-300 ppm for tap water.
But holy hell, you should see what my RO filters look like after just a year's time. Yuck! Dark brown and slimy. I'll have to take a photo and share it with you next time. I just imagine what that would cumulatively do to my internal system over the next 30-40 years. And god only knows what chemicals are in there.
My RO system filters my water down to 4-5 ppm. Tastes amazing and makes great tasting coffee. I just take plenty of supplements and eat lots of veggies, fruit, and lean meats.