This link
http://www.dafyomi.co.il/chulin/insites/ch-dt-130.htm
says we don't give the kohen the zeroa, etc nowadays in Eretz Yisroel based on a kohen not being muchzak. Who gave this opinion and why is this so? What do they do with the zeroa, lechyayaim and kaiva if not to the kohen? Do they burn it? Let it rot? Can you ask or speak to a schochet in Israel? The Rambam in Hilchos Bikurim chapter 9 gives this as Halacha as does the Shulchan Aruch YD 61.
Zvi Shapiro, Long Beach, CA
It is the opinion of the Minchas Chinuch (see Teshuvos v'Hanhagos, IV, #346) that we are exempt today from Matanos Kehunah because Kohanim today are not Muchzak. [Please write back and clarify what you mean by "why is this so".]
Unlike Terumah, these Matanos Kehunah have no special Kedushah and therefore they do not need to be destroyed. They either have to be given to a Kohen or they can be kept and eaten by the owner of the animal.
I was not able to speak to a Shochet here in Israel, but I was able to speak to Rav Moshe Sternbuch who is in charge of all of the Shochtim of the Eida Charedis. He said that many years ago, the Eida had a policy of giving a certain Kohen a grush for each animal they slaughtered. In other words they would be Mezakeh him with the Matanos Kehunah and then effectively buy them back from him for a grush.
They do not do that any longer. Now what they do is buy the animal from Marbek (the large meat corporation here in Israel) only after the Shechitah - everything depends on who owned the animal at the time of Shechitah - and rely on the assumption that someone in that corporation is either a Kohen or a non-Jew - an animal that is owned in partnership with either a Kohen or a non-Jew is exempt from Matanos Kehunah - or that the owners of the corporation each own less than a Perutah in each of the animals, or that a corporation does not have proper ownership of the animal that obligates them in Matanos Kehunah (see Teshuvos v'Hanhagos, op. cit.).
Kol Tuv,
Yonasan Sigler
I think I heard that the Chasam Sofer used to especially shecht an animal once a year and give Matanos Kehunah to a Kohen
Kol Tuv,
Meir Eliezer Bergman, Manchester UK
Indeed, the Chasam Sofer writes about himself (Responsa YD 301) that his custom was to Shecht an animal every Yom Tov and give Matanos Kehunah. He says (ibid) that Rav Yehonasan Eibshitz also gave Matanos Kehunah. The Pe'as ha'Shulchan (3:39) says that the Vilna Gaon performed the Mitzvah of Matanos Kehunah and even made a Berachah on the Mitzvah (however, the Vilna Gaon himself in the Biur ha'Gra (OC 8) says that one does not make a Berachah on the Mitzvah).
Kol Tuv,
Yonasan Sigler