Is my hero left-handed or right-handed?

In WoW and other games you see some difference in using an speed weapon in your left weapon slot and your right weapon slot.

Is there any difference in Diablo 3? Should I use the X weapon in slot Y?

The other answer is not always correct. Some skills use only the main hand weapon damage: see here or here. These tend to be skills that fire infrequently, possibly with long cooldowns. If those skills fire less frequently than the slower weapon, attack speed is irrelevant. Only weapon damage matters. For those skills, you are better off having the weapon with the higher damage in your main hand -- even if it has a lower damage per second (DPS).

Note: in most cases, the other answer should be correct. Most skills alternate hands. Only particular builds are affected at all and they are only impacted when the relevant skill is used.

No, there is no difference in Diablo III.

http://www.osirisguide.com/diablo-3-dual-wield-guide/ gives you an explanation of how dual wielding works with weapons of differing speeds.

However... there is a bug/odd behaviour affecting the character info display (and potentially actual DPS, though that is unconfirmed) in D3 currently. http://us.battle.net/d3/en/forum/topic/4015201923 provides information on this, but one (non-Blizzard) source explains it as:

however when dual wielding take note that MH and OH are not as simple like they are in WoW, the primary vs secondary weapon IIRC is based on equip order.

when dual weidling different APS weapons, you will get different final APS depending on which one you equipped last. afaik the last one you equip becomes secondary no matter if you put it in the char screen slot right OR left.

(this was to some degree exploitable by dual claw assassins in D2 LoD)