For those who have trouble believing that breaches of the
"noahide laws" are punished with beheading / decapitation,
the Jewish Encyclopedia online seems to be clear that the
punishment for a breach of any of the "laws" is decapitation,
as below. So kind of them to use the "least painful" means
of execution.

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LAWS, NOACHIAN:

By : Isidore Singer   Julius H. Greenstone  

ARTICLE HEADINGS:
  The Seven Laws.
  Laws Before Sinai.
  Procedure.
  Special Exceptions.

The Seven Laws.

Laws which were supposed by the Rabbis to have been binding upon
mankind at large even before the revelation at Sinai, and which are
still binding upon non-Jews. The term Noachian indicates the
universality of these ordinances, since the whole human race was
supposed to be descended from the three sons of Noah, who alone
survived the Flood. Although only those laws which are found in the
earlier chapters of the Pentateuch, before the record of the
revelation at Sinai, should, it would seem, be binding upon all
mankind, yet the Rabbis discarded some and, by hermeneutic rules or in
accordance with some tradition (see Judah ha-Levi, "Cuzari," iii. 73),
introduced others which are not found there. Basing their views on the
passage in Gen. ii. 16, they declared that the following six
commandments were enjoined upon Adam: (1) not to worship idols; (2)
not to blaspheme the name of God; (3) to establish courts of justice;
(4) not to kill; (5) not to commit adultery; and (6) not to rob (Gen.
R. xvi. 9, xxiv. 5; Cant. R. i. 16; comp. Seder 'Olam Rabbah, ed.
Ratner, ch. v. and notes, Wilna, 1897; Maimonides, "Yad," Melakim, ix.
1). A seventh commandment was added after the Flood - not to eat flesh
that had been cut from a living animal (Gen. ix. 4). Thus,the Talmud
frequently speaks of "the seven laws of the sons of Noah," which were
regarded as obligatory upon all mankind, in contradistinction to those
that were binding upon Israelites only (Tosef., 'Ab. Zarah, ix. 4;
Sanh. 56a et seq.).

While many additions were made to these laws by some of the
tannaimue.g., the prohibitions against eating the blood of a living
animal, against the emasculation of animals, against sorcery, against
pairing animals of different species, and against grafting trees of
different kinds (ib. 56b) - so that in one place thirty Noachian laws
are mentioned (H.ul. 92a; comp. Yer. 'Ab. Zarah ii. 1), the prevalent
opinion in the Talmud is that there are only seven laws which are
binding upon all mankind. In another baraita (Tanna debe Menasseh) 
the seven Noachian prohibitions are enumerated as applying to the
following: (1) idolatry, (2) adultery, (3) murder, (4) robbery, (5)
eating of a limb cut from a living animal, (6) the emasculation of
animals, (7) the pairing of animals of different species (Sanh. 56b).

Laws Before Sinai.

With regard to the other laws which are mentioned in the Book of
Genesis and which were not included among the Noachian laws, as, for
instance, circumcision and the prohibition against eating of the
"sinew that shrank," the Rabbis laid down the following principle:
"Every law that was enjoined upon the Noachid - and was repeated at
Sinai is meant to apply both to Israelites and to non-Israelites; laws
that were enjoined upon the Noachidae and were not repeated at Sinai
apply to Israelites only" (Sanh. 59a; R. Jose ben H.anina; comp.
Bacher, "Ag. Pal. Amor." i. 430 and note). By this principle a number
of the pre-Sinaitic laws were excluded from the Noachian laws,
although it required a great deal of speculative reasoning to make
this principle apply to all cases (Sanh. 59b).

In the elaboration of these seven Noachian laws, and in assigning
punishments for their transgression, the Rabbis are sometimes more
lenient and sometimes more rigorous with Noachidae than with
Israelites. 

************************** My Emphasis ***************************
With but a few exceptions, the punishment meted out to a Noachid 
for the transgression of any of the seven laws is decapitation, 
the least painful of the four modes of execution of criminals 
(see Capital Punishment).
******************************************************************

The many formalities of procedure essential when the accused is an 
Israelite need not be observed in the case of the Noachid. The latter 
may be convicted on the testimony of one witness, even on that of 
relatives, but not on that of a woman. He need have had no warning
("hatra'ah") from the witnesses; and a single judge may pass sentence 
on him (ib. 57a, b; "Yad," l.c. ix. 14). With regard to idolatry, he 
can be found guilty only if he worshiped an idol in the regular form 
in which that particular deity is usually worshiped; while in the case 
of blasphemy he may be found guilty, even when he has blasphemed with 
one of the attributes of God's name - an action which, if committed by 
an Israelite, would not be regarded as criminal (ib. 56b; see Blasphemy).

[... ....]

Full article at ... "JewishEncyclopedia"
http://www.jewishencyclopedia.com/view.jsp?artid=113&letter=L


Cheers, Steve..

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