Black and green tea do contain caffeine, but quite a bit less: http://www.mayoclinic.org/healthy-lifestyle/nutrition-and-he...

It depends a bit on the specific preparation but black tea contains at most half as much caffeine as brewed coffee (per liter) and green tea is again half as much.

Thus an easy way to reduce your caffeine intake is to replace a cup coffee by a cup of the same size of black tea and maybe later green tea (you are not going to drink twice as much liquid and are not drastically cutting your caffeine intake).

Sure, OP was saying "lots and lots of tea", though :-)

Of course. Switching to lots of it and working down from there.

The English term, "theine" (not to be confused with theanine, an amino acid analogue also found in tea), is not commonly used anymore. It's occasionally used by cranks who insist that it's actually a different and preferable compound to caffeine.