How can I hold conversations with my Pokemon?

I have actually managed this.

Unfortunately, I haven't managed to find a site to source, but it is possible. Playing with an Eevee that I have three hearts with, I called its nickname just playing around. When it seemed to hear, I told it it was adorable and it gave that little sound of excitement like when you pet it, though I did not see any hearts or notes. I continued to give various praise with the same results.

I have yet to try this with other Pokémon, but its actually rather fun to just talk to the Eevee.

Hope this helps at least some.

Page 368, Official Kalos Region Guide says;

You can also talk or sing into your Nintendo 3DS microphone to entertain your Pokemon through verbal communication.

So I take my headphones, turn on classic radio, turn volume at 100%, put headphones on 3DS mic and finally turn on Pokemon Amie.

My Mewtwo aren't such fan of classics but it gived him +2 enjoyment after about 30 second session. No more even after tuning Beetles and listening all "Here come the Sun" so this must be max. Didn't tryied dubstep through.

So it kinda works.

My Banette (five hearts of affection) tends to enjoy hearing her name, but not much else. Yes, my Banette is female, and I've tried saying multiple things to my Banette (mostly praise). Though she does seem quite happy, no notes or hearts will pop up after I speak to her. Though she would not respond when she had no hearts of affection. I do believe it has to do with the level of affection, and I'm going to get Banette's level of affection to six hearts just to try this out. Hopefully it will help some people. My Banette doesn't really respond to her Pokémon name, and her nickname is in Japanese (I received her over the GTS) but I can't pronounce Japanese.

With Eevee (I've used both male and female), the adorable Pokémon loves praise and hearing it's nickname. I've tried playing a few songs for it from the radio, of multiple genres, and Eevee seems to enjoy most songs.

I tried speaking to Empoleon and Blaziken as well, and neither of the two responded.

I also experimented with hacked vs. Not hacked Pokémon. There is little difference. I then tried experimenting with Pokémon that I caught and Pokemon I traded, to see if the OT makes a difference. It does, pokemon of my own tend to hear their nicknames and respond faster than Pokémon I received from trading. I want to see if legendaries make a difference too, but haven't gotten to that yet.

Update: My Banette (six hearts of affection) now responds to her English Pokémon name.

I still have not checked the legendaries to find a difference.