Does the prohibition against consuming new grain ("Chodosh") before the second day of Passover (Leveticus 23:14) apply only in the Land of Israel or in the Diaspora as well? (Sources appreciated)
Those who hold it is forbidden in the Diaspora:
Rif, Rosh, Rambam, Rav Avraham Ben HaRambam, Semag, Semak, Ittur, Mordechai, Raavyah, Haghot Maymoniot, Ritva, Eshkol, Kolbo, Tashbetz, Sefer HaChinuch, Or Zarua, Rabbeinu Yerucham, Meiri, Terumat HaDeshen, Piskei Tosfot, Haghot Ashiri, Tur, Bach, Beit Yosef, Rama.
Those who hold it is permitted in the Diaspora:
Rabbeinu Baruch, Riva.
I'm keeping this to pre-Shulchan Aruch. Please edit in any other rishonim you can find.
Sources: Last Biur Halacha on OC 489; Tur YD 293; Chinuch 303; Toldot Adam VeChava 5:4:45; Kolbo 56; Beit HaBechira Challah 1; Semak 217; Shut Rav Avraham Ben HaRambam 85; Terumat HaDeshen 191; Eshkol 159b; Maamar Chametz LeRashbatz 140
The prohibition to eat "new" grain (grain which was harvested after the day of the omer offering, which is on Day 2 of Pesach) does apply in the Diaspora, according to most poskim.
Wheat often sits in silos for a year or two after harvesting, before it is made into flour. grains like barley are dried and sealed in plastic bags, and they can sit in a warehouse or on the shelf for years.
The Rema brings a double doubt: maybe the grain product you now possess is old, and even if it was harvested this year, maybe it took root before Pesach.
R' Moshe Feinstein says that one may rely on this Rema, but it is better to be strict, if possible.
Until recently, most kosher consumers in the Diaspora were not able to be strict.
Today, one can obtain a list of barcodes for bags of flour in the supermarket (at least in the United States). Thus, it is possible to find out exactly when any given bag of flour was milled. If it was milled before Pesach of this year, obviously the grain took root before Pesach, and it is yoshon.
I couldn't find a specific website for barcodes, but this site will help anyone who wants to start ensuring that all grain in their own home is yoshon http://www.star-k.org/kashrus/kk-yoshon-prepyoshon.htm
Certainy one may rely on the Rema, as cited by R' Moshe Feinstein, and eat in the home of any shomer kashrus, or any reliably certified kosher restaurant, regardless of their zeal with barcode checking on bags of flour and dried barley.
However, the issur of chadash is a Torah prohibition. If one has sufficient storage space at home to purchase all grain products before the first chadash could possibly hit the stores (if memory serves, you have from pesach until at least Rosh Hashannah), that is a praiseworthy practice.
According to S"A Y"D 293 it applies everywhere and the Rama doesn't disagree. However, the Ashkenazim accepted the Bah (on the Tur) to be Mekil out of Eres Yisrael on gentiles grain.
It's a machlokes Tanna'im, Amora'im, Rishonim, Achronim and contemporary poskim. The original machlokes is how to interpret "bchol moshvoseichem" in the Torah. Most of the standard rishonim holds it does apply mdoraysa in chutz l'aretz (Rambam, Rif, Rosh) and this is how the shulchan aruch paskens. Most people have been lenient on this issue for a long time, with various kulos suggested (see Rema, Bach, Taz, Aruch haShulchan).