It's actually quite hard to say for certain for a variety of reasons. To begin with, most of the time the family members just aren't written down, so it's impossible to know for sure where their ancestors are from. Alongside that, even if we do know their family members, intermarriage between Armenian, Greek, and other Anatolian aristocracies meant that often times people were a mix. You also have to contend with the fact that many people were settled and resettled, so just because we know you're from Thrace doesn't mean you're Thracian. This compounds with the fact that generally if you were a native Greek speaker and weren't a member of some noteworthy tribe of Barbarians you would be simply written down as "Greek." Finally, there's the fact that many of the dynasties weren't dynasties in the European sense but were just members of an extended family (cousins, sons in law, etc).

Also, to make clearer something /u/alriclofgar made note of, I'm talking about ethnicity in the modern sense. In the Roman World, you were either Roman (Latin or Greek Speaker (ish) which is why you see so many people who aren't Greek (Armenians everywhereeeeeeee) as emperor, along with following roman cultural practices) or not. The simple answer is "Roman," but I'll go more into their ancestry.

Leonid Dynasty

Source: I'm actually not aware of what the primary sources are for this dynasty, so I'm just going off of the very cursory knowledge that I've gained from various survey texts I've read and online sources.

Leo was of Thracian descent (Hence his moniker Leo the Thracian), but was heavily romanized. The Thracians weren't particularly Greek in ethnicity, and would have spoken a Vulgar Latin dialect related to the modern endangered language Aromanian, so this just gets recorded as "Latin." He would have been seen as Roman, however. His son would have been of similar stock.

Zeno- Zeno was an Isaurian, or a "barbarian" tribe that inhabited the Isaurian mountains near Cilicia. The Isaurians had been subjects of the Empire for quite some time and were seen as not too bad of barbarians, although he certainly wasn't Roman or Greek. I'm not aware of the language he spoke, but due to the differences in the names of Isaurians and native Greeks, he probably didn't speak Latin or Greek.

Anastasius I- He was of Illyrian stock, so from the region of the north west balkans. Would have spoken Latin and at that point was seen as fully Roman (Which he had to be, due to the fact that the Hippodrome supposedly chanted "Give Rome a Roman Emperor," (as a way to highlight that Leo was not 'Roman') to Ariadne, the daughter of Leo I and wife of Zeno).

Justinian Dynasty

Source: Procopius

Justin and Justinian: I'm going to just group them when I can. Justin and Justinian were both born in what is modern Serbia and were of Illyrian or possibly Thracian descent, but like Anastasius were thoroughly Romanized. They were peasants, however. They would have spoken native Latin, and Justinian was actually the first to legislate in Greek (although his Greek was supposedly pretty bad)

Justin II- Would have been of similar stock due to being born of the sister of Justinian. I'm not quite sure what his father's blood was, but I would assume that Justinian would have married her to a prominent Latin or Greek family of Constantinople, so he was likely either half greek or full Roman.

Tiberius II - Was born in Thrace, most likely to Latin speakers. Other than that, I can't seem to find any more information from the sources I have on hand.

Maurice - Was born in Cappadocia, which again means we're not QUITE sure of what his background is. According to the Oxford Dictionary of Byzantium, it's plausible that he was either a Cappadocian Greek, Romanized Armenian, or some mix of the two (Cappadocia was a traditionally Greek area, but there were also native Anatolians and Armenians in the area, so it was sort of a melting pot). Regardless, he was a native Greek speaker and would have been seen as Greek.

Phokas According to the lovely A History of Byzantine State and Society by Warren Treadgold, he was a native Greek speaker of Thrace. This is ambiguous, as many Greeks lived in Thrace as did many Romanized Thracians. This again highlights how nobody really cared about your ethnic background as long as you were thoroughly Romanized, so the difference doesn't particularly matter. However, since his native tongue was Greek, it's likely he was of native Greek descent.

Heraclian Dynasty-

Source: Theophanes the Confessor and Warren Treadgold,

Heraclius was originally from Cappadocia and was of Armenian descent. So he'd have looked pretty similar to the Greeks of the Empire. This is, thankfully, where adoptions start to become not as frequent, so there will be less writing since most of the time people were related to each other by blood from now on.

Constantine III - Eldest son of Heraclius and his wife Fabia Eudokia. She was born in Africa although was most likely the child of a Greek landowner. So Constantine was still a mix of Armenian, Greek, and maybe some Berber/North African.

Heraklonas. Mother was Martina, who was the daughter of Heraclius's sister, and his Father was Heraclius. We're not sure who Martina's father was other than that his name was Martinus, but it's likely it was some Greek speaker in Constantinople.

Constans II- Son of Constantine III and his wife Gregoria, who was the daughter of Niketas, a cousin of Heraclius I. So as such, she would have been of similar Greek-speaking chimera descent.

Constantine IV - Son of Constans II and Fausta, who was the daughter of Valentinus, a general of Armenian descent, potentially from the Arascid dynasty (the guys who ruled both Armenia and the Parthian Emire). So, would have been half Greekish and half Armenian.

Justinian II- Son of Constantine IV and Anastasia, whose ancestry is not known. She likely was a native Greek speaker of Constantinople or some Armenian aristocrat.

Usurpers of Justinian II

During first overthrow-

Leontios - Born in Isauria, but at this point the "Isaurians," as a people were probably no longer recognizable. Would have spoken Greek and probably looked more Arabic than the other Emperors, but again, this isn't really important since he would have been thoroughly Romanized.

Tiberius III - Was actually, according to the sources, a Germanic person whose original name was Apsimar. As a high naval chief, he would have had to be able to speak Greek fluently, so it's likely that his family was a resettled German family that had been Romanized.

2nd Overthrow

Philippokis- His father was Nikephoros, a patrician of Armenian descent that lived in Western Anatolia. His mother is not recorded, so it's unknown if he was mixed with any Greek, but he most likely was.

Anastasius II - I can't find anything about his background. I generally default to "most likely Greek" in these circumstances.

Tiberius III- Same as Anastasius, so we're not sure.

Isaurian Dynasty

Yay actual dynasty again.

Source: More of Theophanes and Treadgold.

Leo III - He was of "Isaurian," descent (whatever that means at this point) but was a native Greek speaker that had been resettled in Thrace. Isaurian can also be seen as "Syrian," at this point, so he might have been more middle eastern, but we can't be sure.

Constantine V - Son of Leo III and Maria, whose ancestry and family is not known. Most likely Greek.

Leo IV - Son of Constantine V and Tzitzak, a Khazar princess. The Khazars were a turkic people from the Steppes of Asia, so she would have looked much different than the Greeks around her. For modern Turkic peoples, look at the Uyghurs (I don't say the "Turks," because it's likely that they're mostly descended from the native Greeks of Anatolia). His nickname "The Khazar," probably confirms that he looked different than the other Greeks.

Constantine VI- Son of Leo IV and Irene. Irene was a relative of a powerful Greek family in Athens. As such, she was certainly very "Greek." She also becomes Empress.

Non-Dynastic mostly

Source: More Theophanes and Treadgold, but we can finally also rely on the Madrid Skylitzes, which my translation of is incredibly thorough and well annotated, so this information is likely a bit more accurate.

Nikephoros I - No note of his family background that's reputable. From Selukia Sidera in what is now central southern Turkey, so he was a native Greek.

Staurakios - Son of Nikephoros I. No mention is made of his mother, likely Native Greek.

Michael I - Most likely native Greek, but otherwise Skylitzes makes no mention of his ancestry. Was married to the daughter of Nikephoros I and had multiple children who were all castrated. One of them Niketas/Ignatious became Patriarch.

Leo V- His nickname "The Armenian," should help you figure out his family. His father was Bardas, a patrician of Armeniain descent. My book makes mention of the fact that some later sources also claim he was a descendant of some Assyrians, but I can't speak to the veracity of this claim.

Amorian Dynasty

Source: I'm going to almost exclusively using John Wortley's translation of the Madrid Skylitzes at this point, especially because Wortley notes the differences between it and the work of Theophanes at this point and also cites Treadgold and other sources where needed.

Michael II - No mention of his family, but he was born into a Greek speaking land owning family in Phrygia. He was also a member of Athinganoi, a sect of Christians that followed many Jewish practices. The fact that Skylitzes differentiates between the Athinganoi and Greeks/Armenians may mean that they were seen as "different," than Greeks, or just that a part of the Greek/Roman identity was orthodox Christianity.

To be continued when I'm done letting an Organic Chemistry exam violate me.