Generally, whenever we perform a Misva that applies infrequently, we recite the Beracha of "She’heheyanu" in addition to the Beracha recited over the Misva. Thus, for example, on every Yom Tob we recite "She’heheyanu" over the occasion of the holiday and for the special Misvot that apply. Similarly, this Beracha is recited before one blows the Shofar on Rosh Hashanah and before one reads the Megila on Purim.
This gives rise to the question of why we do not recite "She’heheyanu" before counting the Omer for the first time each year. This Misva, too, occurs infrequently, and it would thus seem appropriate to mark the occasion of Sefirat Ha’omer by reciting "She’heheyanu."
There are two famous answers given to this question. The Meiri (Rabbi Menahem Meiri, 13th century) explains (listen to audio recording for precise citation) that the "She’heheyanu" recited on Shabuot covers Sefirat Ha’omer, as well. In the times of the Mikdash, the counting of the Omer served to count the days toward the special sacrificial offering brought from the newly harvested wheat on Shabuot. (This sacrifice was called the "Korban Sheteh Ha’lehem.") The Misva of counting was therefore intrinsically bound to the Misva of the Shabuot sacrifice. As such, no separate Beracha of "She’heheyanu" was recited for Sefirat Ha’omer, as it was covered by the Beracha recited on Shabuot. And even though nowadays we do not bring this special sacrifice due to the absence of the Bet Ha’mikdash, we nevertheless do not recite "She’heheyanu" just as this Beracha was not recited during the times of the Temple.
A different answer was proposed by Rabbenu Zerahya Halevi (the "Ba’al Ha’ma’or,"
1125-1186; listen to audio recording for precise citation). He claimed that the Beracha of "She’heheyanu" is recited only for occasions that involve a degree of enjoyment or from which we derive some kind of benefit. We recite this Beracha before shaking the Lulab, for example, as the Lulab is part of our joy and celebration on Sukkot. This Beracha is similarly recited before the sounding of the Shofar, which arouses divine compassion. The reading of the Megila clearly brings us joy over the miracles that occurred for our ancestors, and thus it, too, is preceded by the Beracha of "She’heheyanu." A father recites this Beracha at a Pidyon Ha’ben, when his son reaches the age of thirty days, at which point he is presumed healthy (as opposed to the first thirty days after birth, during which the child cannot yet be presumed capable of surviving). All these occasions are accompanied by an element of joy, and thus warrant the recitation of "She’heheyanu." Sefirat Ha’omer, however, as the Ba’al Ha’ma’or describes, evokes feelings of grief, as it brings to mind our inability to offer the sacrifices associated with the Omer period. We can only count the Omer to commemorate what took place during the time of the Temple, as we cannot offer the special sacrifices on Pesah and Shabuot. For this reason, it is customary to recite a brief prayer after counting the Omer for the restoration of the Temple. Hence, the Ba’al Ha’ma’or writes, it would be inappropriate to recite "She’heheyanu" over this Misva, which we are able to perform only as a commemorative act and thus evokes feelings of grief and sorrow, rather than joy and festivity.
