Ukraine did not immediately comment on Moscow's claims.
In a statement, the Russian defence ministry said its forces had "liberated" the settlements of Popiv Yar in the eastern Donetsk region, Degtiarne in the northeast Kharkiv region and Kamianske in the southern Zaporizhzhia region.
Kamianske, on the banks of the Dnipro river, was home to around 2,000 people before the conflict.
Degtiarne is a tiny hamlet near the Russian border, but lies in an area of the front line that Moscow's forces had not penetrated since the early months of its offensive.
Popiv Yar is a small village south of the Ukrainian city of Kramatorsk.
The Russian military accelerated its advances for a third consecutive month in June.
Its territorial gains for that month were the biggest since November last year, according to an AFP analysis of data from the US-based Institute for the Study of War (ISW).
US President Donald Trump has given Moscow "50 days" from Monday to reach a peace deal or face bruising sanctions, but Ukraine says the Kremlin has no interest in ending the fighting and will press on regardless.
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