Gemara

36b (Mishnah): Any Mitzvah that is not dependent on land applies also in Chutz La'aretz, except for Orlah and Kil'ayim;

R. Eliezer says, even Chadash (eating new grain before the Omer is offered).

38b (Mishnah): Chadash is forbidden mid'Oraisa in all places.

Menachos 68b: Rav Papa and Rav Huna brei d'Rav Yehoshua would eat Chodosh on the night after the 16th. They forbid Chodosh in Chutz la'Aretz only mid'Rabanan, so we are lenient about a Safek (which day was Rosh Chodesh).

Rabanan of Rav Ashi's academy ate from the morning of the 17th. They forbid in Chutz la'Aretz mid'Oraisa. (We are stringent about the Safek.) Ravina's father did not eat until the next night (he forbids the entire day of the 16th mid'Oraisa.)

Rishonim

Rif and Rosh (15a and 1:62): The Halachah follows R. Eliezer, for a Stam (anonymous) Mishnah (38b) supports him.

Rambam (Hilchos Ma'achalos Asuros 10:2): One who eats Chadash before the Omer is lashed mid'Oraisa, in Eretz Yisrael and Chutz la'Aretz.

Teshuvas Rosh (2:1): R. Baruch permits even what grows in a Yisrael's Reshus here in Chutz la'Aretz, far from Eretz Yisrael. Chachamim did not decree here, just like they decreed about Ma'aser only near Eretz Yisrael. This is astounding! Even if he exempts grain of Nochrim, the Torah forbids a Yisrael's grain! There is no substance to his proofs. R. Yitzchak and Sefer ha'Mitzvos say that we drink barley beer and eat rye of Nochriml, for we rely on the majority, which is planted before the Omer. Avi ha'Ezri adds that some of the barley is from the year before.

Rosh (ibid.): Surely, this applies in most years. This year, the Nochrim's Inuy (Easter) was shortly before Pesach, and the land was too dry to plow. Only a small minority planted before Pesach. I did not forbid, lest people not heed me. I tell questioners to ask farmers. It is permitted only if most was planted before the Omer. Two or three times in my youth, their Inuy was shortly before Pesach. My Rebbeyim were stringent, but did not publicly forbid. It is better that people be Shogeg, and zealots will refrain. I saw in the name of Maharam (me'Rotenburg) 'some say that we drink beer due to a Sefek-Sefeka (two doubts): perhaps it is not Chadash. And even if it is Chadash, perhaps it took root before the Omer.

(Rosh): Regarding mixtures, we do not permit a Sefek-Sefeka she'Yesh Lo Matirim (it will be totally permitted later, after Nisan 16). Here, we permit (in normal years), since there is no Chezkas Isur.

Terumas ha'Deshen (191): The Or Zaru'a says that Chadash is only mid'Rabanan in Chutz la'Aretz. One should be stringent in a year when they could not plant before Pesach, for then it is a single Safek.

Poskim

Shulchan Aruch (YD 293:2): The Isur of Chadash applies in Eretz Yisrael and Chutz la'Aretz, both to grain of Yisrael and of Nochrim.

Rema (3): We permit Stam grain after Pesach due to a Sefek-Sefeka: perhaps it is not from this year, and even if it is, perhaps it took root before the Omer. Where they surely plant oats and barley after Pesach, one must be stringent, unless most comes from elsewhere where they plant before Pesach. If winter extends after Pesach, and in the entire region they plant after Pesach, one must be stringent. Where most of the food and drink is from these grains, one should not rule stringently to others, for it is better that they be Shogeg than Mezid.

Taz (4): Almost all Chachmei Yisrael are lenient. We cannot say that it is better that they be Shogeg! Even if we explain the Yerushalmi like the Bach (that grain of Nochrim is permitted), we cannot oppose all the Poskim! Perhaps we can justify those who are lenient. Another Stam Mishnah (Orlah 3:9) forbids in Chutz la'Aretz. We do not know which Stam Mishnah was taught last, so we are stringent. In our lands, we drink mostly beer of barley and rye. In She'as ha'Dechak (pressed circumstances), we may rely on the first Tana, since the Gemara did not rule explicitly. Also in Nidah 9b, Rebbi once ruled like R. Eliezer. B'She'as ha'Dechak, he relied on R. Eliezer, for the Halachah was not decided like either opinion. L'Chatchilah, we rule like the majority. B'She'as ha'Dechak, one may rely on R. Eliezer. Also in Gitin 19a, b'She'as ha'Dechak we rely on an individual's lenient opinion. All the more so here we may be lenient, for there are reasons to say that the Halachah is like the first Tana. The stringent Poskim lived in warm climates, where Chadash was rare. In cold lands where winter extends until Pesach, we may be lenient. The Maharshal cites R. Avigdor to say that Chadash does not apply to grain of Nochrim; Bahag concurs. Perhaps they hold like the first Tana.

Rebuttal #1 (Nekudas ha'Kesef): Rabanan of Rav Ashi's academy and Ravina's father forbid Chodosh in Chutz la'Aretz mid'Oraisa. The Sugya in Kidushin connotes like this. In Nidah and Gitin we were lenient b'She'as ha'Dechak only because there was no concern for an Isur mid'Oraisa.

Rebuttal #2 (Gra 2): If the Halachah was fixed, we may not follow the other opinion b'She'as ha'Dechak. The Halachah was fixed here and in Menachos.

Be'er ha'Golah (7): Great Chachamim of old and recently were lenient. They rely on the Yerushalmi that says that when Yisrael entered Eretz Yisrael, they ate Matzah from grain bought from Nochri merchants. Even though Rabanan of Rav Ashi's academy and Ravina forbid Chodosh mid'Oraisa, and we do not follow the Yerushalmi against the Bavli, She'as ha'Dechak is different.

Rebuttal (Gra 2): This is a great mistake. In the Yerushalmi (Chalah 2:1 (10b)), R. Yonah says that the Aseh to eat Matzah overrode the Isur of Chadash. R. Yosi holds that Aseh Docheh Lo Sa'aseh only if they are written together. R. Yosi holds that they ate Matzah from grain of Nochrim, like R. Yishmael (unlike R. Eliezer). The Bavli holds like R. Yonah. Rosh Hashanah 13a proves that the Halachah follows R. Eliezer. Rather, people are lenient because grain is planted after Pesach, like the Rema says.

Magen Avraham (OC 489:17): The Rosh says that "b'Chol Moshvoseichem" teaches that Chadash applies everywhere. Perhaps it forbids grain of Eretz Yisrael in Chutz la'Aretz! This is how we can defend R. Baruch's opinion and the universal practice. A Ba'al Nefesh should be stringent when possible.

Mishnah Berurah (489:45): The Rema's leniency due to a Sefek-Sefeka does not apply to barley, oats and rye in Polin, where these are planted after Pesach and are not imported. Still, most people are not careful. Some say that b'She'as ha'Dechak we rely on the minority of Rishonim who forbid in Chutz la'Aretz only mid'Rabanan, or only near Eretz Yisrael, or permit grain of Nochrim. We cannot protest against those who are lenient, but a Ba'al Nefesh should be stringent as much as possible, for many great Rishonim forbid mid'Oraisa in every case. One should be careful even about wheat imported from Russia, for most of it is Chadash.

Bi'ur Halachah (DH Af): One should not be totally lenient just because it is hard to observe all stringencies, e.g. beer and whiskey and absorptions. One can find wheat seeded in Cheshvan. Some Acharonim are lenient about beer. Also, even in winter perhaps it is from old malt. Many Acharonim are stringent, but one who is lenient has whom to rely on. The Or Zaru'a, Terumas ha'Deshen and Piskei Tosfos were lenient in Chutz la'Aretz only about a Safek, for they hold that it is mid'Rabanan.

Kaf ha'Chayim (111): One must be careful that Matzas Mitzvah not be Chadash, lest it be a Mitzvah ha'Ba'ah b'Aveirah.

Note: This is because one can find old grain. If not, Aseh Docheh Lo Sa'aseh!