Grading: This hadith is sahih, and this chain is hasan on account of Muhammad bin Salih (Hamdan al-Hamadhani) and his shaykh Abd al-Samad bin Hassan, since both are saduq and they are corroborated.
Technical note / Defect ('illah): Although 'Ata' bin al-Sa'ib suffered from mental confusion (ikhtilat), Sufyan al-Thawri heard from him before the confusion.
Important point: Al-Nasa'i, al-Bayhaqi, Ibn al-Salah, Ibn Hajar and others preferred the mawquf narration, while the vagueness over the Companion's name does no harm.
Reference / Source: We have given the details of this in our notes on "Sharh Mushkil al-Athar" (5973).

Reference / Source: Imam al-Tirmidhi (960) narrated it as marfu' by way of Jarir bin Abd al-Hamid from 'Ata'.
Corroborating chains and supporting reports: Although Jarir heard from him after the confusion, he is corroborated by Sufyan al-Thawri and Sufyan bin 'Uyaynah, who heard from him before the confusion.
Important point: Imam al-Tirmidhi says that it is known as marfu' only through the hadith of 'Ata', but we say that it is also narrated as marfu' from Sa'id bin Jubayr with a sahih chain (no. 3093).

Reference / Source: Abd al-Razzaq (9791) narrated it by way of Ja'far bin Sulayman from Ibn 'Abbas (may Allah be pleased with him) as mawquf.
Criticism of a narrator: Ja'far bin Sulayman is a Basran, and the Basrans' narration from 'Ata' dates from after his confusion.

Reference / Source: Imam al-Nasa'i narrated it in "al-Kubra" (3931) by way of Ibrahim bin Maysarah from Tawus from Ibn 'Abbas (may Allah be pleased with him) as mawquf: "Circumambulating the House is prayer, so speak little."

Variation in the chain: Al-Hasan bin Muslim went against Ibrahim bin Maysarah, narrating it as marfu' and leaving the Companion's name unspecified.
Reference / Source: Ahmad and al-Nasa'i narrated it by way of Ibn Jurayj from al-Hasan bin Muslim from Tawus.
Important point: Al-Hafiz Ibn Hajar says that the apparent view is that the unnamed Companion here is Ibn 'Abbas (may Allah be pleased with him) himself, and even if it is someone else, the Companion being unnamed does no harm.

Details of the narration: Hanzalah bin Abi Sufyan narrated it through Tawus from Ibn 'Umar (may Allah be pleased with him) as mawquf, which al-Nasa'i recorded at (2923).
Important point: Imam al-Daraqutni says that the view of those who said "Ibn 'Umar (may Allah be pleased with him)" is the more likely (correct) one.

Note / Clarification: Shaykh al-Albani's (may Allah have mercy on him) excellent discussion of the takhrij of this hadith can be seen in "Irwa' al-Ghalil" (121).

Important point: See further the hadiths after this one, and nos. (3093, 3095).