An Ashkenzaic custom is to not eat nuts on Rosh Hashanah, and some say right through till Yom Kippur. A couple of reasons are given for this. The Maharil says that nuts increase the saliva in the mouth and may interfere with the prayers we say during the high holidays. Since during these days we are praying more than usual, some refrain from nuts that could interfere with that.
The deeper Jewish sources say, that it is connected to the numerical value of the Hebrew word for nut, Egoz, which has the same numerical value as "Chet" (sin), 17. Since we want to stay far away from sin as possible, some stay away from any foods, which may even conjure the idea of sin in our lives. Although the number 17 is also the gematria for "tov" (good), some respond that it is better to not even allow sin to be even mixed in with good during these days.
More important than these customs of course, is to pray fervently and to enjoy all of our meals on these very special and holy days!