Gemara

104b (Agra, the father-in-law of R. Aba - Beraisa): Fowl and cheese may be eaten wantonly.

(Agra): This means that one need not wash his hands or clean his mouth in between.

Rav Yitzchak brei d'Rav Mesharshiya visited Rav Ashi. He ate cheese and then beef without washing his hands in between.

Onlookers: Agra's Beraisa permits eating fowl and cheese wantonly. This implies that beef and cheese are forbidden (without washing)!

Rav Yitzchak: That applies at night. During the day, one can see that his hands are clean.

(Beraisa - Beis Shamai): One cleans his mouth (between meat and milk);

Beis Hillel say, he washes.

Beis Shamai say that he must clean his mouth. The same applies to washing. Beis Hillel say that he must wash. The same applies to cleaning his mouth.

Each said a different law. They do not argue.

(R. Zeira): Bread must be used to clean the mouth.

It must be wheat bread, but not barley bread.

The bread must be cold. If it is hot, it will stick to his palate.

It must be soft, and not hard.

The Halachah is, anything may be used, except for flour, dates and Yerek (vegetables).

R. Asi asked how long must one wait between cheese and meat. R. Yochanan answered that he need not wait at all.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 9:26): If one ate cheese or milk, he may eat meat immediately afterwards. He must wash his hands and clean his mouth in between.

Kesef Mishneh: Rashi, Tosfos and the Rosh say that 'washing' is risning the mouth. The Rambam holds that it is washing the hands. A support is that the Gemara discussed eating fowl without washing his hands and cleaning the mouth. It did not mention rinsing the mouth. Regarding Rav Yitzchak, the Gemara discussed only washing his hands. The custom is to fulfill both opinions, and wash (and clean) the mouth and wash the hands.

Radvaz: Another support is that the Beraisa of Beis Shamai and Beis Hillel is brought after the episode with Rav Yitzchak. However, this is difficult, for the Gemara suggested that washing or cleaning helps. How does washing the hands help for cheese stuck in the mouth? This requires investigation.

Rambam (27): One who eats fowl after cheese or milk need not clean the mouth or wash his hands.

Rosh (8:5): The Rif brought Agra's Beraisa, which permits eating fowl and cheese wantonly. This shows that he holds that milk and fowl is mid'Rabanan, like R. Akiva. Agra permits fowl and cheese without washing the hands and cleaning his mouth in between. The same applies to Chayos and milk. He mentioned fowl, for it is more common. R. Tam says that he discussed fowl, for it does not cling to the hands, teeth and gums.

Ran (37a DH v'Hu): Agra explained that if one ate cheese after fowl and now wants to eat the other, he need not wash and clean. For beef, one must clean, lest some of what he ate clong to his gums. The Rambam holds that Agra is lenient only for fowl after cheese, but not for cheese after fowl. The episode with Rav Yitzchak proves this. They brought to him meat after cheese, and ate without washing. They challenged this from the Beraisa, from which we infer that one must wash for beef after cheese. If the Beraisa discussed also cheese after fowl, this is not difficult. Perhaps it mentions fowl to teach that one need not wash for cheese after fowl, but one must wash for cheese after beef, but indeed, one need not wash even for beef after cheese! Rather, this shows that the Beraisa discusses only fowl after cheese. Below, the Gemara forbids meat after cheese. Stam 'meat' includes fowl. The Rambam says that Agra permits even cheese after fowl. His words 'fowl and cheese' connote like this. Even so, we challenged Rav Yitzchak, for if beef after cheese were permitted, why did Agra teach that the Chidush of cheese after fowl? He should have taught that sometimes we are lenient about beef and cheese, about which there is a Torah Isur, e.g. when one eats beef after cheese! Since Agra taught only fowl, this implies that for this we always permit, i.e. even cheese after fowl, but there is no Heter for beef and cheese, even for beef after cheese. Afterwards the Gemara forbids meat after cheese. Stam meat refers to beef. We hold that the Torah does not forbid Chayos with milk, therefore they are like fowl.

Rashba (Toras ha'Bayis ha'Katzar 3:4 86a, brought Beis Yosef YD 89 DH Achal): Some permit cheese after fowl or Chayos immediately without washing and cleaning, but they err. The Rambam holds that this is like eating after meat of a Behemah. However, if one ate cheese, he may eat fowl or Chayos without washing and cleaning.

Rosh (ibid.): Beis Shamai say that he must clean his mouth. The same applies to washing. Beis Hillel say that he must wash his hands. The same applies to cleaning. They do not argue. Beis Shamai said first to clean, and did not mention washing. Beis Hillel said that washing is better than cleaning. Beis Shamai admitted. Both are needed. If either of them sufficed, what did Beis Hillel add? Therefore, one must clean and rinse the mouth with water between cheese and meat, in addition to washing the hands if it is night.

Gra (YD 89:9): The Rashba explains that it suffices to clean or rinse. If not, why did Beis Shamai and Beis Hillel and Agra mention only one?

Poskim

Rema (YD 89:1): Some say that one need not wait six hours after meat before eating milk. However, he must say Birkas ha'Mazon in between.

Mishbetzos Zahav (3): The Taz says that surely one may not say Birkas ha'Mazon in order to eat cheese afterwards. The custom is not to eat cheese for six hours. Some are lenient (to eat meat) after soft cheese, and rely on cleaning and rinsing. Even the stringent opinion blesses (on Shavu'os) Birkas ha'Mazon on the dairy meal, and prepares to eat meat. Since it is a stringency, we do not add to it.

Shach (6): If he was not eating bread, he must say a Berachah Acharonah in between.

Sifsei Da'as (6): The Acharonim say that the same applies to one who is stringent not to eat meat after cheese, based on the Zohar. He must say a Berachah Acharonah after milk before eating meat. On Shavu'os the custom is to eat dairy, bless Birkas ha'Mazon, and then eat meat.

Shulchan Aruch (2): If one ate cheese, he may eat meat immediately afterwards, as long as he checks his hands that none of the cheese stuck to them. If it is night and he cannot check them well, he must wash them. He must clean his mouth and rinse it. Cleaning is through chewing bread or anything he wants, except for flour, dates or vegetables, since they stick to the Chanichim (above where one swallows, close to the teeth). Afterwards he rinses his mouth with water or wine. This is to eat meat of a Behemah or Chayah afterwards. If he will eat fowl after cheese, he need not clean or wash.

Beis Yosef (DH Achal): The Rambam explains that Agra discusses eating fowl after cheese. We would not be lenient about cheese after fowl. The Ramban holds that it permits even cheese after fowl. The Ran says that we are similarly lenient about Chayos with milk, for also they are only mid'Rabanan. The Rambam equates Chayos to Behemos, even though the same applies to Chayah and milk is only mid'Rabanan. Perhaps his reason is because meat of Chayos resembles meat of Behemos.

Shach (9): R. Yerucham says that if one has a good lamp, he need not wash his hands at night. The Tur brings from R. Peretz that one should wash his hands even during the day, for sometimes the cheese is fatty and sticks to dirt on the hands, and one is unaware. The Acharonim hold like this; this is the custom.

Shach (11): One may clean or rinse the mouth first. The order does not matter.

Pischei Teshuvah (5): The Pri To'ar says that it is not enough to chew. He must swallow to be considered cleaning. In any case one should not spit out, due to wasting food.

Pischei Teshuvah (6) Keneses ha'Gedolah brings an opinion that all Peros do not help, but he concludes that Peros other than dates help.

Rema: Some are stringent even for meat after cheese. This is the custom. If the cheese is hard, we do not eat after it even fowl, just like we do not eat cheese after meat. Some are lenient. One should not protest, just they should clean and wash the hands. However, it is good to be stringent.

Taz (4): Isur v'Heter says that the stringent opinion requires waiting six hours after cheese. Cheese is hard if it is six months old or has holes. Really, we forbid only milk after meat. The taste of meat remains in his mouth, so we decree lest he eat meat with milk. One should be stringent about meat after cheese due to Midas Chasidus. I disagree. The Rambam forbids cheese after meat due to meat left between his teeth. He is not concerned for taste left in one's mouth. There is no concern after eating cheese. Even meat between the teeth would not be called meat if not for the verse "ha'Basar Od Bein Shineihem" (105a)! If it is due to taste, letter of the law we must be stringent after eating wormy cheese. However, after old cheese that is not wormy, that was not made from milk curdled in a stomach, rather, from regular milk that was dried, or after butter, it suffices to clean (the mouth) and remove particles from the teeth and rinse the mouth and hands, unless one wants to be extra stringent.

Shach (15): Stam, if cheese is six months old, it is considered hard.

Note: Nowadays, through chemicals, some hard cheeses are as if they were aged for six months, but no company actually ages cheese for six months. Poskim argue about whether the stringency of hard cheese applies to them.

Gra (11): The Zohar is stringent. We do not say that it is against the Gemara, for also they (Chachamim of the Gemara) used to be stringent.