Different variants of this name of village there are several in Bohemia. Exactly Okrouhlé (alternative old name Okrouhlí) is part of the village Svatojanský Újezd, the district Jičín. Parish is in Lázně Bělohrad. But that's not it.
But almost at 99%, I would say that it can be Okrouhlá near town Milevsko. From nearby village Opařany were Bartůněk family as well and from Okrouhlá also family Boreš, what emigrated to the United States in the 19th century too. It is a village situated 4 km south of Milevsko, the district Pisek, South Bohemian Region.
Finally, after quite a long search for the right village, I found in the parish registers, as follows:
Collection of registers of South Bohemia, Parish: Sepekov, Book: 4 (N 1833-1845) | Picture: 73, Page 71http://digi.ceskearchivy.cz/cs/7165/73
Born on 19 July 1839, Okrouhlá Nr.19, Joseph Bartunek, Father John Bartunek, crofters from Okrouhlá, son of Vaclav Bartunek, in retirement from Okrouhlá and wife Catherine Smrt daughter of a farmer from Něžovice. Mother Catherine daughter of Antonin Prochazka, a day laborer from Okrouhlá and (wife) Veronika Rybak, daughter of a farmer from Okrouhlá.
Midwife Marie Boreš, a peasant woman from Okrouhlá Nr.10
Godparents: Jan Mach, ¼ (= smaller) farmer from Okrouhlá Nr.2, Catherine Pesek peasant woman from Okrouhlá Nr.4
Collection of registers of South Bohemia, Parish: Sepekov, Book: 5 (N 1845-1851) | Picture: 5, Page 3http://digi.ceskearchivy.cz/cs/7166/5
Born on 2. September 1845, in Okrouhlá Nr.22, Anna Boreš, father Joseph Boreš, crofter from Okrouhlá Nr.22, domain of Milevsko, son of Francis Boreš, ½ farmer from Okrouhlá Nr.20 and mother Mary née Toman from Rukáveč. Mother Rosalie Matuška, daughter of the deceased John Matuska, a farmer from Okrouhlá Nr.11, domain of Milevsko, and mother Anna nee Kofron from Milevsko.
Midwife Marie Boreš, ½ peasant woman from Okrouhlá Nr.20, untried (=without midwife exams)
Godparents: Anna Němec crofter woman from Okrouhlá Nr.17, Frank Matuška, farmer from Okrouhlá nr.11
Collection of registers of South Bohemia, Parish: Sepekov, Book: 14 (O 1862-1895) | Picture: 24, Page 22http://digi.ceskearchivy.cz/cs/7175/24
Wedding: 22 November 1864, Okrouhlá Nr.19/22,
Groom: Joseph Bartunek, crofters from Okrouhlá Nr.19, son of John Bartunek, crofters also from Okrouhlá Nr.19, mother Catherine nee Procháska, daughter of Anthony Prochaska a day laborer on Okrouhlá, both parents Catholic religion. Groom 25 years old, single.
Bride: Boreš Anna, the daughter of Catholic parents, Josef Boreš, crofters from Okrouhlá Nr.22 and mother Rosalie Matuška, daughter of the deceased John Matuska, a farmer from Okrouhlá nr.11. Bride 19 years, single.
Witnesses: Jan Boreš, farmer from Okrouhlá Nr.20, Joseph Rybák, a farmer from Okrouhlá Nr.1
Official notes and father's permission to wedding for underage brides.
Collection of registers of South Bohemia, Parish: Sepekov, Book: 7 (N 1857-1871) | Picture: 129, Page 128http://digi.ceskearchivy.cz/cs/7168/129
Born on 6. March 1866, in Okrouhlá Nr.19, Joseph Bartunek, father Joseph Bartunek, crofters from Okrouhlá Nr.19, son of John Bartunek, crofters also from Okrouhlá Nr.19 and mother Catherine nee Prochaska, daughter of Anthony Prochaska a day laborer from Okrouhlá. Mother Anna Boršová (=Boreš), daughter of Joseph Borš (=Boreš), crofters from Okrouhlá Nr.22 and mother Rosalie Matuška, daughter of the deceased John Matuška, a farmer from Okrouhlá Nr.11.
Midwife Rosalie Boršová, crofter woman from Okrouhlá Nr.22, untried (=without midwife exams)
Godparents: Joseph Boreš, crofters from Okrouhlá Nr.32, (Testes:) Rosalie Boršová his wife
You can further research your family on-line on http://digi.ceskearchivy.cz/DA?lang=en
Bartunek family went to Cleveland, lived in Cuyahoga, Ohio - is that right? And you know that it is possible to find at www.mundia.com
a family pedigree of Bartunek - added by “mgbergstr” nickname?http://www.mundia.com/us/Person/15653016/2003904993
Have a nice Christmas