Cork taint doesn't only come from corks, it turns out; it can come from barrels. One way that it forms (in part) is when chlorine bleach is used to clean corks (or barrels).
Wikipedia says, "Chlorinated phenols can form chemically when hypochlorous acid (HOCl-, one of the active forms of chlorine) or chlorine radicals come in contact with wood (untreated, such as barrels or pallets.) The use of chlorine or other halogen-based sanitizing agents is being phased out of the wine industry in favor of peroxide or peracetic acid preparations."
Much tap water contains chlorine or chloramine, so don't clean out your barrels with untreated tap water.
Depending on whether your water is treated with chlorine or chloramine you may take a different approach to getting rid of that in the water (as opposed to buying gallons and gallons of distilled water). Chlorine and chloramine require different filters or amount of time boiling the water or time to leave it to fizz off.
A little bit of research gives widely different answers as to how long you'd have to boil water to eliminate chloramine (that's what's in San Francisco's drinking water). The answers are everywhere from 20 minutes to 2 hours to 2 days of boiling. Carbon filters also remove chloramine, but they have to be really good/fresh filters. Some detailed information from a brewing perspective is here.
This was first brought to my attention by Carl Sutton of Sutton Cellars. I asked him what a good cleaner for barrels would be and he recommended Proxycarb. Some research tells me that has the same active ingredient (Sodium Percarbonate) as OxyClean (though I don't know if OxyClean is food-safe so you should probably buy it from a wine/beer store).
Have fun with your barrel aged cocktails, and remember to avoid chlorine when cleaning them out.