If I believe in Yeshua, won't that mean I'm no longer Jewish?

Not at all!

One's Jewishness is determined by birth (i.e., whether or not one's parents are Jewish), not by one's religious beliefs. Even if one did not want to be Jewish anymore, he could not do anything to change it. In addition, Yeshua was born of a Jewish mother (see the following New Testament reference: Galatians 4:4) and lived a Jewish lifestyle. He consistently followed the Jewish traditions and taught others to do so (see Matthew 5:17-19; 23:3).

The early followers of Yeshua (called disciples) worshipped daily in the Jerusalem Temple (see Acts 2:46; 3:1). Also, the B'rit Chadashah (New Testament) was written by Jewish authors (except possibly Luke) to explain Jewish teachings to a Jewish audience. The Jewishness of the New Testament is clearly reflected in a recent translation entitled "The Jewish New Testament".

Believing in Yeshua, the Jewish Messiah, is a very Jewish thing to do.