I do, so do many others.
Firstly there are different branches of the Eastern Church, I am of the Byzantine/Greek Rite (those Catholic Churches who ritually resemble the Greek and Russian Eastern Orthodox), so I can only speak of us.
Because Adoration of the Holy Mystery is considered by many to be a “latinization” and “foreign to the Eastern Church” (although however true most old Eastern Catholics you come across will recount fond memories of doing such things as children, not very foreign in my opinion) you usually will not see Adoration, however it is still common to have it in some parts of Eastern Europe.
Honestly, I don’t get the issue about “latinization” if we are all Catholic, but the main thing I really don’t understand is why was it “genuine” enough in Europe but not in the US! In other words, if in Eastern Europe, where the Eastern Church comes from, they do “it”, why not here?!

I have to tell you, the Latin church can ban it, and I would still find my way to it!
Depends on where you are at, I pray it as do many others but in some parts outside of our countries of origin (America in particular) it is officially discouraged because it is viewed as being “Latin”.
Hope that helps