Are Christians allowed to practice Eastern martial arts? I know some people, Christians, who study Eastern martial arts. They explain this by saying that they do not practice the spiritual aspects of karate or kung-fu; but they use only the physical component of martial arts (i.e., they study it as gymnastics). 

The spiritual component of Eastern martial arts can be substituted with prayer to God.
In general, yes, it is allowed.  However, I want to point out this verse:

1 Corinthians 10:23 (NIV)
“I have the right to do anything,” you say—but not everything is beneficial. “I have the right to do anything”—but not everything is constructive.

This verse comes right before a passage talking about eating meat sacrificed to idols.  In general, this is allowed for us.  However we have to be careful to make sure that what we are doing does not take us away from God, but brings us closer.

1 Corinthians 10:31
So whether you eat or drink or whatever you do, do it all for the glory of God.

If we follow God and do everything for his Glory, we're doing things the right way.
Summary
If you're feeling like God is calling you to not participate in any activity, you should avoid that activity in order to glorify God.  Otherwise, feel free to participate in that activity so long as you do it for God's glory.