I request random songs until I get too many repeats, then I find the list of his songs online and start requesting them one by one. Which is exactly what I did this time around again.
But I don't think many people are going to be hurting for music this time around (or having to follow this procedure), because you can buy songs from the Nooklings and even order them straight from the catalog.
I'm very probably going to be giving away songs as fast as I can I get them in the mailbox when NA/EU comes out to help get everyone up to speed, because I already have all but one of them.
I make a list and cross it off as I request them in order (except for the songs I like to play in my house like Go K.K. Rider and Imperial K.K.) until completed. Sometimes I use other characters to speed up the process since there are so many songs.