The source you are looking for is the Rema's (Rabbi Moshe Isserles) commentary for Ashkenazim on the Shulchan Aruch. Yoreh Deah 89:1
It starts with the opinion of R' Yosef Karo (Tzefardim usually follow him.):
One who eats meat, even of a wild animal or fowl, does not eat
cheese afterwards until he waits six hours. Even if he waits
that period, if he has meat between his teeth he has to remove it. One
who chews food for a child has to wait.
The Rema then comments by saying that some hold you do not need to wait any amount of time at all from meat to milk. Then he says the Ashkenazi custom he is familiar with in his locale is 1 hour. Finally at the end, he does endorse the 6 hour time frame used by R' Yosef Karo.
"...There are those that says that he doesn't have to wait six
hours, but rather immediately if he finishes the (meat) meal and
says the concluding blessing, it is permissible after wiping and
rinsing his mouth (Tosafot Hullin 105a - "At the next meal", Mordechai
chapter 25, Haga'ot Ashri, Haga'ot Maimoniot chapter 9 of forbidden
foods, and Ravya). The simple custom in our countries is to wait
after eating meat one hour and to eat cheese afterwards, but you
have to say the concluding blessing after the meat (HaAruch, Haga'ot
Shaarei Dura) because then it's like a new meal and permissible to eat
according to the lenient view. But with no blessing, waiting alone
does no good. It doesn't matter if you waited before the blessing or
afterwards (his own reasoning, from the Mahari, as opposed to the
Issur v'Heter). If he finds meat between his teeth after the hour, he
has to pull it out (his own reasoning, from the Ran above). And there
are those that say not to say the concluding blessing in order to eat
cheese (Aruch in the name of Maharach) but we're not careful about
this. And some are careful to wait six hours after eating meat
before eating cheese, and it's proper to do so."
So the Shulchan Aruch does not mention a concept of 3 hours. It is either nothing, 1 hour, or six hours.
However, other authorities say or suggest other amounts of time in between 0 - 6 hours. If you just wanted to know if there is an opinion that you only need 0 or 1 hour for Ashkenazim, then yes, there is such an opinion. Your concept of 3 hours certainly fulfills the more lenient opinions which require less waiting time.
Origin (sources) of the three hour wait can be found discussed here:
Sources for 3 hour time limit (not Rb. Yerucham)
As always, this answer is for learning purposes. If someone has a true question about how many hours they should or should not be waiting between meat and milk, they should consult their personal Orthodox Rabbi for guidance on what is right for them.