There certainly were. From the Second Punic War where Rome first started using foreign troops, Rome utilised soldiers from specific places in specific roles due to their specialist skills. The first foreign troops the Romans began using in the Second Punic War were Syracusan light infantry, Cretan archers, Numidian infantry and especially cavalry and later Balearic slingers. As the Romans stopped using Italian cavalry around the time of the Social War due to manpower shortages, pretty much all cavalry used by the army were foreigners. In his Gallic campaigns Caesar used legionaries as heavy infantry, Cretan archers and Balearic slingers as skirmishers, and Gauls and later Germans as cavalry. Occasionally he and others would also use friendly Gauls as infantry. This sort of mix was fairly typical of the period, and the region from which these auxiliary units were recruited were generally associated with a particular kind of troops. German cavalry were particularly well regarded, but Gauls began serving quite commonly as both infantry and cavalry in the Late Republic, which is also around the time the Romans started forming actual legions of foreigners, such as the Pontic Legion.

With the annexation of Gaul and the regions of the Rhine and Danube, in the early empire had access to very good raw material for soldiers in the form of Germanic barbarian groups. These men served in large numbers as auxiliaries, formed into their own units but equipped similarly to legionaries with a similar, if not more "combat oriented" function than the proper legions, who retained their unique expertise in engineering and siege warfare. German units formed from particular groups such as the Batavians had excellent reputations for virtus, Germans were trusted to guard certain Emperors due to their mercenary loyalty, and into the Late Empire Germanic soldiers became hugely represented as not only auxiliary heavy infantry but proper legionaries. Of course, with the Edict of Caracalla in 212, with all free males in the Empire becoming citizens, the distinction between Italian or Italian-style legions and German auxiliaries blurred a great deal, but one of the Later Empire flairs will have to explore this in more detail for you.

With the expansion of Roman power in the East and their ensuing conflicts with fairly formidable eastern powers such as the Parthian and later the Sassanids, the Romans began to use 'eastern' troops in large numbers, once again as specialist auxiliaries. Syrians and other eastern groups were particularly known as excellent archers, and unlike the mercenary Cretan and Balearic skirmishers the Romans formed regular auxiliary units of Syrian and eastern archers and posted them with heavy infantry units all over the Empire. Rome adopted similar practices with horse archer and cataphract-style cavalry units, forming their own direct equivalents to the elite cavalry of the Persians as a response to the effectiveness of mass cavalry in that part of the world.

The trusty Italians continued to serve as legionaries and dominate the officer class of the army, though as I've mentioned while the Romans had begun farming out the battlefield roles of light infantry and cavalry of various types to mercenaries and sometimes auxiliaries from the time of the Second Punic War, the distinction between legionaries and heavy-infantry auxiliaries, who were often German, blurred over time with local recruitment and eventually disappeared entirely.

The Romans evidently respected tactical effectiveness and specialist skills, and were fairly expansive in their idea of citizenship, so were perfectly happy to lead soldiers from other cultures in their army. Of course, Roman moral stereotypes still dominated the Italian officer class (in particular disciplina continued to be considered a specifically Roman trait, though barbarians were accepted to have possessed ample virtus to serve as heavy infantry), but essentially the Romans recruited and incorporated auxiliaries into their armies from specific regions, organised into specialist units, based on their peculiar military skills, from the Mid Republic onward.