You do NOT have to take a full-round action to get more than one ATTACK per round. End of story. Anyone who says differently needs to go back and read some rules.
Here are but a FEW examples:
Haste. - Haste gives you an extra partial action. You may attack with this partial action or you may dance the funky chicken. This is irregardless of any other previous actions in the round. So you could move your full move (move equivelent)... then attack (standard)... then attack again (extra haste partial).
ATTACK of oppurtunity - This is a special case... but it's STILL an attack. You can make an attack of oppurtunity on a creature who provokes you irregardless of what type of action you took/are taking/will take. With Combat Reflexes, you can make MANY attacks of oppurtunity.
Expert Tactician - When an opponent is denied his dexterity bonus, you make an extra attack on him irregardless of whether you are using a full round action. A common use is for rogues to Bluff (move equivilent), then attack as a sneak attack (bonus attack from expert tactician) and then attack (standard attack).
Improved Trip - If you successfully trip an opponent, you make a free attack on them. In other words... move up to the person (move equivilent), trip attack (standard attack) and then, provided the trip was successful) extra attack (improved trip extra attack).
NOTE - Trip is a validate attack option to use during an attack of oppurtunity which if sucecssful, would give 2 attacks (AoO as a trip, then regular attack from Improved Trip) without taking a full round action.
So what do the above examples prove?! NOTHING! Except that the "no multiple attacks without a full round action" is a worthless argument.
That said... here's my take.
Speed weapons use the "haste" action, so if you were hasted, you couldn't utilize both at the same time. It's like trying to combine two bonuses of the same type (armor bonuses don't stack, enhancement bonuses don't stack, etc.).
HOWEVER, I would allow both weapons of speed to use the SAME haste action and go off simultaneously, thus invoking all the penalties associated with two-weapon fighting.
Why? For one reason.... if someone dumps two feats into Two-weapon fighting and Ambidexterity, there has to be SOME pay-off.... even if it doesn't come until you can afford to buy (or have someone make) Speed weapons... which shouldn't be until at least level 12... when the wizard is teleporting all over the place and disintegrating things.
I would NOT allow a munckin player to try to claim that they go off differently and therefore you can effectively fighting with two weapons without the feats.
But hey... that's just my opinion.