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.