There are a number of answers brought down and several have already been listed. A few additional answers are as follows:
In Sefer Hamanhig (Hilchos Pesach, 64) by Rav Avraham ben Natan HaYarchi, he notes that Shehechiyanu by definition is only recited when the action being done elicits a degree of Hana'ah (benefit) such as with Pidyon Haben, Shofar, Lulav, Succah etc. However, Sefiras Haomer doesn't create any such pleasure. As such:
ספירת העומר אין בו לא מעשה ולא הנאה למצוה
The counting of the Omer is neither an act nor a pleasure for the mitzvah
Another approach is that sefiras ha'omer is wholly dependent on the festival of Pesach and is therefore included in the shehechiyanu that one makes on Leil HaSeder (Seder Night).
This opinion is adopted by Rav Tzidkiyahu ben Avraham Anav who explains in his Shibolei Haleket HaShalem:
שלא נהגו לברך שהחיינו לפי שזמן ספירה תלוי בקביעות בפסח כמה דתימר וספרתם לכם ממחרת השבת לכן נראה שאין מברכין עליו זמן ודי לו בברכת זמן של יום טוב עצמו
We are not accustomed to make Shehechiyanu, since the time of Sefira is dependent on the permanence/fixed nature of Pesach like we say, "And you shall count for yourselves from the day after Shabbos (i.e. after the first day of Pesach)..." - therefore it appears that we don't make a bracha on the time (of sefira) and it is enough for it (i.e. the bracha of shehechiyanu) to be included with the bracha of the time of Yom Tov itself (i.e. included with Pesach).
This view is also upheld by the Meiri on Pesachim 7a - s.v. 'Sha'alu' who asserts:
שאני אומר שספירת העומר נפטר הוא בזמן של יום טוב
Since I say that Sefiras Haomer 'departs' at the time of Yom Tov (i.e. Pesach)