All this if for a campaign world I'm brewing now that is low on magic. The most commonly used classes are martial (barbarian, fighter, ranger, rogue) and magic is mainly used by the bad guys, and only full spell caster characters (cleric, druid, sorcerer, warlock, and wizard, not sure how to handle the bard yet...) which are very rare.
Edit: I've always been a huge fan of pulp sword and sorcery and I want to make a world based on that. Conan and Red Sonja comics and art by Ken Kelly, Frank Frazetta, Boris Vallejo and others is what inspired me.
Edit 2: Really looking forward for the new
Conan RPG coming this summer!
I'm listening to the original soundtrack of Conan the Barbarian while writing this so you can get the idea...
Honestly, 5e may not be the way to go. Think about it: The PH has 38 sub-class options. About 30 of them cast spells. The Champion, Battlemaster, Berserker, Thief and Assassin are the only ones that are fairly unambiguously non-magical. No class lacks a magic-using archetype. Non-magical 'healing' (HD) is inadequate, and inaccessible in combat.
It's a very high-magic and magic-dependent game.
3.5 has numerous fighter builds, barbarian, rogue, knight, scout, numerous PrCs, the Marshal f/Battlesystem - and, heck, the first 3 levels of ranger. It's obscure, but there's an 'inherent bonuses' option to reduce the need for magic-items to maintain balance as you level. The only downside is the need for potions or UMD+WoCLW for healing. Pathfinder adds more non-caster classes, including it's own take on the Marshal.
4e offers you something like 6 builds each of Fighter, Ranger, Rogue and Warlord. Essentials adds the Knight, Slayer, and Thief and post-Essentials the Berserker (you have to be pretty careful to avoid primal powers, but you can do it) and, I guess, Executioner (it might have a magical 'shadow' power or two, I forget). If you ignore the Essentials classes, they're all reasonably balanced. You can use inherent bonuses to remove magic items from the expected advancement. And, non-magical healing, especially with a Warlord, is quite sufficient.
13th Age has a Fighter, Ranger, Rogue, and (in 13TW) Commander, with modest variety among them, and, like 4e, has adequate non-magical healing, including an in-combat 'Rally' option.
Finally, if you can dig up the d20 Iron Heroes, it's prettymuch exactly what you're looking for - mighty-thewed barbarians, big weapons, manly combat, and not too much magic. It has 10 PC classes - /one/ of them is a caster, and it's optional. Oh, and, 'reserves' - a form of non-magical 'healing' used in a few d20 games to take up some of the slack from the loss of magical healing.