Fighter:Race

Player's Handbook
Dragonborn — A bonus to strength and a racial ability that lets you mark multiple enemies make this one of the best PHB races. The to-hit bonus when he's blooded is icing on the cake.

Dwarf — Bonuses to two important secondary abilities (Con and Wis), along with second wind as a minor action, make this a solid choice. The Dwarven Weapon Training feat is a great way to get access to superior weapons and have a damage bonus on top of it. A particularly good choice for battleragers, hammer users, and battleraging hammer users.

Eladrin — The dexterity bonus can be good for some weapon users; with their weapon training feat, longsword and spear users benefit in particular from this race. Teleporting into a new position isn't bad, either.

Elf — With bonus to secondary abilities (Dex and Wis) and a good racial to keep from missing, these are to heavy blade users what dwarves are to hammer users. Might do particularly well as tempest; pit fighter tempests benefit a great deal from high wisdom. Use a scimitar to get the most mileage out of a high dexterity score.

Half-Elf — The constitution may be helpful for hammers and axes, but the charisma won't help at all. At least fighters have a lot of options for useful Dilettante powers.

Halfling — Like the half elf, one useful bonus for some weapon users. Generally being limited to smaller weapons is bad for fighters, but they'll do well as tempests; like elves they could use scimitars to make use of a high dexterity.

Human — Can be good at any type of fighter. With martial power out they shouldn't have a problem finding three useful fighter at wills.

Tiefling — Ability bonuses aren't helpful, though a flaming weapon can make up for it.

Forgotten Realms Player's Guide
Drow — With only the dexterity bonus being useful, these may seem like poor fighters, but blinding all the enemies surrounding you so that they either have to waste their turn attacking you at a -5 penalty or take an attack of opportunity moving away (use the feat to increase the area of effect so just shifting away won't work) is a good enough ability to make up for it in some cases.

Genasi — The strength bonus is good, but the real benefit is in the encounter powers; an earthsoul can knock adjacent enemies prone as a minor action to make an even sticker fighter. Other types of genasi can up damage or mobility.

Dragon Magazine
Minotaur (Dragon 369) — Great hammer user.. Good battleragers, too.

Gnoll (Dragon 367) — Their stats let them be good with either dexterity or constitution based weapons. Their good mobility and bonuses to damage if you're bloodied and/or ganging up on a target with two others are handy, especially as a tempest.

Warforged (Dragon 364) — Again, great for axes, hammers, and battleragers. A racial encounter power allows them to gain temporary hit points. If they are bloodied, this same power will also give them real hit points.

Races appearing in the following sources were not intended for use by player characters. Most of these have very limited racial options. Consult with your DM before creating a character using one of these races.

Monster Manual
Bugbear — With a bonus to strength and dexterity, plus having oversized weapons, these are good dexterity-based weapon users. Their racial ability is icing on the cake.

Doppleganger — No useful ability bonuses and a non-combat racial. Leave this to rogues and warlocks.

Githyanki — Constitution is useful with the right weapon, and the racial abilities aren't bad.

Githzerai — Same modifiers as an elf; they're better if you want a high initiative to charge out ahead of your party. Racial ability is pretty good, too.

Gnome — While fade away is fun, it doesn't help a defender, and neither do the stat bonuses.

Goblin — Dexterity bonus and a power to reposition most rounds makes this decent with the right kind of weapon; you'd probably want to be a tempest.

Hobgoblin — The constitution bonus is good with the right weapon, racial power is so-so.

Kobold — The kobold probably won't get much use out of its constitution bonus, being small and thus restricted from the most damaging hammers. A dexterity weapon using tempest could work. Shifty is a great power to get you where you want to be.

Orc — Excellent stats for axes or hammers, plus can charge from far away, plus can heal himself once per encounter. A great choice, particularly for battleragers.

Minotaur — With bonuses to Strength and Constitution, as well as the ability to wield oversized weapons, minotaurs can make devastating axe or hammer fighters. Their racial encounter isn't bad either.

Shadar-Kai — Similar to the eladrin, without the benefit of racial weapons.

Longtooth Shifter — A strength bonus and wisdom bonus, plus a racial ability to get extra damage and regeneration when you're bloodied, make these good as any type of fighter and particularly good as pit fighters. A tempest longtooth pitfighter is especially frightening when he's been bloodied.

Razorclaw Shifter — Neither the ability mods or the racial power are as good as the longtooth, but they can still work for dexterity based weapon users.

Manual of the Planes
Bladeling — A dragonborn, with its strength bonus, is probably a better choice, but if you want to use a dexterity based weapon, have a decent wisdom, and have an area-effect minor power for marking, this is where to go. Their racial may be better than a dragonborn, hitting around you instead of a block in front of you, and can be upgraded a great deal with a single feat.