This is a great question, but I am finding it to be very difficult to answer. The 4th and 3rd millennia's are proving to be very challenging in terms of finding historical documentation relating to specific cities (as this timeframe is often considered Pre-History). That being said, it does seem that archaeology has provided us with some clues as to how often flooding occurred. There are several large clay deposits, which can be dated to several periods, varying from the Late Ubaid period (in Ur) to the Early Dynastic III period (Kiš). Devastating floods were not unheard-of during the first half of the third millennium BCE. The rivers sustained life in Mesopotamia, but they also destroyed it by frequent flooding.
I believe the best place to start would be archaeological evidence. For this we begin with Sir Charles Leonard Woolley. Woolley was a British archaeologist best known for his excavations at Ur in Mesopotamia. He is considered to have been one of the first "modern" archaeologists, and was knighted in 1935 for his contributions to the discipline of archaeology. Woolley does appear to try to tie together ancient "stories/traditions" with his archaeological finds. For the purposes of my answer I am staying away from this portion of his research as this is where things get very tough in regards to accuracy and factual evidence. Regardless it does appear that the city of Ur did experience a dramatic flood around 3100 BC during the Uruk Period (ca. 4000 to 3100 BC). The clay deposits found by Woolley were 3.75 meter thick. This is to say that post flood at least (likely much more) 3.75 meters of sediment were found in the area. For that much sediment to be laid down you could expect that said flood consisted of a lot of water. However, this flood (much like the others that follow) does not seem to have been a "world" flood. In this particular case no archealogical evidence of flood residue, of the same time period, was found just 23 kilometers (12 kilometers depending on source) from the city of Ur in the Sumerian city of Eridu. Note: Ur was established during the Ubaid Period ca. 6500 to 3800 BC (Sidenotes: Ubadian Culture & meaning of Ubaid).
Woolley's team found evidence of several more floods. The city of Kish (Kiš) which would have been occupied during the Jemdet Nasr Period (3100 -2900 BC) experience large flooding. Also archaeological evidence of flooding was found in the Sumerian city of Shuruppak (Shuruppag/Šuruppak) which would have been occupied during the end of the Early Dynastic I period (c. 2750). As well as, In the city of Uruk during the beginning of the Early Dynastic I Period. Note: This flood was not during the the Uruk Period, but the city of Uruk was still occupied. Lastly Woolley's team also found archealogical evidence for another flood in Kish during the Early Dynastic III Period (c. 2450 BC).
As previously stated "this timeframe is often considered Pre-History" therefore it is very difficult to suggest, aside from using "stories/tradition", that any of the above mentioned floods "wiped" out any one of the cities mentioned above. However, it would be accurate to suggest that due to the archaeological evidence that the flooding was indeed significant. Thus, one could say that possibly a city or cities underwent dramatic change post flooding. One may even suggest that one or more of these floods were reasons for the "changes of power" resulting in "new periods" though this would likely be considered conjecture.
ADDITIONAL INFORMATION NOT ORIGINALLY INCLUDED DUE TO POSSIBLE CONJECTURE:
I find the following passage very interesting, however, I could not find verification outside of Sumerian tradition. And in this case it is very unlikely that any archeological evidence can be found.
The delta could only be made habitable by large-scale irrigation and
flood control, which was managed first by a priestly class and
then by godlike kings. Except for the period 2370-2230 B. c., when the
Sumerian city-states were subdued by the rulers of Akkad , the region
immediately to the north
Map of Ancient Mesopotamia:
As for why the area doesn't flood now or in recent history. It does, Iraq, to this day still experiences flooding.
The Tigris River winds its way from its birthplace in the mountains of
eastern Turkey through Iraq to the Shatt al Arab and the Persian Gulf.
Fed by mountain snow and rainfall, the river is prone to springtime
As for why cities in this area are no longer destroyed by floods, I would have to say that this is a result of geographical surveying and other prior planning in city design. Also beginning in the 1950's Mesopotamian Marshes have undergone planned/designed draining.