I use elements of Eberron as part of my homebrew mashup setting and have in 3.5, Pathfinder 1e, and 5e.
I think 5e is a little better fit than 3.5 for some Eberron themes in a couple of ways.
5e has a couple advantages over 3.5 in doing pulp:
Broader skills with bounded accuracy and a minimum of four skills per character means it is easier for more PCs to do well at pulp action type things. As a DM I found 5e better than 3.5 for running an Indiana Jones style crazy action chase scene in my game.
For pulp action combat things 5e characters are more durable and resilient, they have the same hp as 3.5 but death saves instead of death at -10, healing back from 0 instead of negatives so a 1st level healing spell can always get downed characters back into the action, and more non magic healing to get back to full with short rest hit dice and long rest full heal.
5e combat dynamics is a little more pulp hero action than 3.5. In 5e you don't draw AoOs from charging into melee combat fists flying, only from leaving threatened spaces. You get full attacks even when moving so there is not the 3.5 advantage of drawing in a charging opponent who would only get one attack so you can retaliate with a full attack, just get in there in 5e when you decide you need to fight.
5e is designed for the noir anybody any alignment ethos more than 3.5 is out of the box. Monks don't need to be lawful, barbarians and bards can be lawful, there is also almost no mechanical impact of alignment in 5e. You could say everyone is actually neutral with shades of gray and 5e runs the same with no mechanical adjustments needed. I loved the 3.5 Eberron cleric setup of casters who could be any alignment and 5e core is closer to that than 3.5 core was. While the 5e PH does talk about the central importance of gods for clerics, the 5e DMG tells you that all that matters mechanically is their domain and it is up to the DM how they handle gods if at all and even the 5e PH has the Eberron philosophy clerics as a core sample pantheon.
The only new core class in 5e is warlocks, which were supplemental in 3.5 and are super easy to integrate into most every world. Swap out Rakshasa for fiend patrons and quori or daelkyr work well for great old one patrons.
The things I would expect some friction on would be Sarlona psionics stuff and possibly the handling of artificers. If you were really into the mechanics of their implementation in 3.5 then 5e's real simplified mechanics could be disappointing and a bit jarring. Mostly 5e psionics are just magic powers or spells with psionic flavor, not their own full on variant system with spellpoints and psionic focus and their own full classes the way 3.5's complete psionics was. For artificers the whole xp and gold for crafting system is gone so the 5e artificer class is mechanically different, a partial caster who makes a few item things, very much focused on the flavor of the class concept over the 3.5 mechanics. I've had a warforged artificer PC play in my 5e game and it went really well for us. I find them to have been an odd class in 3.5, 4e, and 5e, but mostly equally odd in each edition.
