Okay, why non-Roman units? We are all fond of the named and numbered legions, they are perfect, unique, awesome. Now let's party with the other guys.
Besides it wouldn't be fair to let the legionaries compete with others, they would overwhelm their opponents anyway.
Why infantry units? I always put my faith in the footmen. Infantry is the only arm where the weapon is the man himself.
They provided the bulk of the armies in the era, they gave the structure of the battles, and the elite units delivered the smashing strike in most cases.
They are powerful, superior, they are the hands of the gods.
So now imagine you have a time machine, and you can go back to the ancient world to explore it and do whatever you want (but visiting Rome is verboten).
Which of these units would you enlist?
eastern heavy infantry (armenia)
armenian legionary (armenia)
bastarnae (bosporus)
sacred band of ba'al (carthage)
elite african infantry (carthage)
poeni heavy phalanx (carthage)
elite pikemen (epeiros)
naked fanatics (gaul)
heavy spear infantry (gaul)
armoured warrior (gaul)
germanic cherusci (germania)
chosen axemen (germania)
dii warriors (getae)
tarabostae swordsmen (getae)
spartan hoplite (greek cities)
armoured hoplite (greek cities)
lusitanian elite shock infantry (iberia)
bear warriors (iberia)
iberian neito guard (iberia)
royal pikemen (macedon)
pergamian hoplites (pergamum)
bronze shield phalanx (pontus)
scythian bonded swordsman (scythia)
steppe warrior (scythia)
basilikon agema (seleucid)
silver shield pikemen (seleucid)
hoplites of syracuse (syracuse)
Consider the following when choosing units:
- salary (recruitment cost / no. of soldiers)
- uniforms: you know, this is what the hottest chicks in town are craving for...
- equipment: you may look supercool in your outfit, but if your weapons are junk, you will be tumbled down in the battle
Personally, I would be a Spartan hoplite of course: the payment is very competitive (1.75 denarii per man), and the gear, both the outfit and the weapons are just superior.
I would also put myself to the test as the member of Ba'al's Sacred Band, and as a Bastarnae (just too sicko to resist).
I have made pictures of all the RS non-Roman elite infantry units to ease your job. 
Take a peep.
Armenia
Bosporan
Carthage
Epeiros
Gaul
Germania
Getae
Greek Cities
Iberia
Macedon
Pergamum
Pontus
Scythia
Seleucid Empire
Syracuse
Note: I have pinched the idea from this thread.