There are many features that Pokemon never brought back in the future games. Here are some of them that I always missed, didn't care about, or just hated, and hopefully the ones that I did miss will come back in Gen 7...
Seasons
Back in Gen 5, seasons were a thing. They cycled through each season depending on the month, and it was fun seeing how the season changed the landscape of Unova. In the winter, snow would pile up, letting you access some new areas while not letting you enter others. Fall brought colorful leaves, summer brought earlier light, and spring brought more rain. In addition, the seasons meant different wild encounters, different Trainer teams, and different Deerling/Sawsbuck forms. It's a shame that they didn't return in Gen 6...
Pokemon Following You
I always wanted to see my Pokemon follow me as a young child while watching the Pokemon anime and seeing Pokemon walking along with their trainers. And Gen 4 brought that. (It was also in Yellow but that was only Pikachu and I hate Pikachu) In D/P/Pt, cute Pokemon could follow you in Amity Square and give you Contest stuff or Berries. In HGSS, that was developed even further so that ANY Pokemon could follow you anywhere outside (some size restrictions indoors, like how reality works). It was fun seeing your Pokemon follow you, and you could even talk to them and see how they were feeling, with different Pokemon feeling differently in different areas (i.e. Grass types in Ice Cave). I know that they would have more sprites to make in 3D due to walking diagonally, but it would not be a waste of time in our beliefs.
Contests/Concerts
Before ORAS came out with its remade Contests, it seemed that it was over for Contests. Gen 5 had turned them into Concerts, and Gen 6 didn't bring either back. I don't care too much for them, but they were a very nice break from Battles, as you could dress your Pokemon up or allow it to use better moves that showed how your Pokemon was better than any of the other competitors, showing that you were the supreme Pokemon Coordinator AND Trainer. Plus you could collect Ribbons! However, whether they return in Gen 7, nobody knows.
Puddles
I don't care too much about these, but they were a cool little way of implementing more wild encounters. Plus, in winter, they would freeze, and you could go sliding on the ice!
Fog
I hated Fog. Worst Weather ever. It was taken out after Gen 4, luckily, but I still feel that it should've been improved rather than just completely thrown in the trash.
Trainer Customization
I really liked customizing my Trainer to make him/her look more like me, but it didn't return in ORAS for the reasons of ORAS being a remake of Ruby and Sapphire and Kalos being based off of France, a place where fashion is more important. Hopefully this one returns in Gen 7...
Underground Mining
I enjoyed going underground in Sinnoh and just mining away to find Shards, Stones, Orbs, Plates, and even Fossils. It seemed fairly rewarding, and getting to make a Secret Base was also pretty fun!
Secret Bases
Having a little base of my very own, which I could decorate with anything I wanted, was something I liked. You could buy Pokedolls or other Decorations, as well as set traps in your base for adventurous Trainers who dared come into your base to get your Flag. It went from RSE to DPPt to ORAS, but who knows if it'll come back?
Rebattleable Gym Leaders
In Gen 4, you could find Gym Leaders waiting for you to battle in a little restaurant area or get their Phone Numbers and battle them, as they bring stronger teams to challenge you once again. In Gen 5, you could fight some older Leaders in the Championships. But not in Gen 6... I really was disappointed by the challenge XY and ORAS provided, and I wanted to fight stronger Gym Leaders (and stronger Elite 4 in XY).
Battle Frontier
In Gens 3 and 4, the Battle Frontier was the way to challenge your skills, as you had to defeat 7/5 different Facilities full of Trainers or Wild Pokemon using your own Pokemon or Rental Pokemon, and then finally defeat the Brain behind the Facility. Then, to get Gold, you'd have to do it all over again! However, Gen 5 turned it into the Battle Subway, and Gen 6 turned it into the Battle Maison. ORAS even teased us by having a sign saying that the Battle Frontier is under construction and an agent NPC saying that Pike Queen Lucy has been recruited. Perhaps Gen 7 will bring this back.
So... what do you miss/hate/don't care about?