Know is a strong word, but there are many, many muslims who claims to be decendents of Muhammad. There is, however, a long tradition of Islamic genealogy, although that needs to be understood as confessional, not historical, even though there is some interest from anthropology, since this genealogy is based mostly in studying the belonging to a tribe and then having that tribe associated with the Quraysh (the tribe of Muhammad).

There are certains titles that are given to those who claim to be descendants, and in some countries (like Egypt) you can get a certificate to prove it. There they are called al'ashraf (plural), and the title itself is sharif or Sayyid (see Ayatollah Sayyid Ali Khamenei, Supreme Leader of Iran).

The House of Hashim are among those who trace their origin to the prophet. Current head is King Abdullah II of Jordan.

There are also non-Muslim who others claim decends from Muhammad, such as the royal baby. So basically Muhammad can be seen as an Islamic version of Genghis Khan, who many claim is the ancestor of like 9% of Eurasian population today (which I don't know if it's true, but that's what some say).

I don't know of any Muslims who trace their origin to Muhammad through any other way than Fatima, but there might be those who say they descend from Zainab. Maybe someone else will be able to fill in there.

Sources and further reading (apart from those linked):

Asfaruddin, Asma, “The Issue of Succession to the Prophet”, The First Muslims. History and Memory. Oxford: One World. 2008. pp. 19-26

Clarke, Morgan, "Closness in the Age of Mechanical Reproduction – Debating Kinship and Biomedicine in Lebanon and the Middle East”, Anthropological Quarterly, Vol. 80, No. 2 (2007) pp. 379-402.

Esposito, John L., Islam: the Straight Path. Fourth edition. New York: Oxford University Press. 2011 (1988)

Lindholm, Charles, “Kinship Structures and Political Authority: The Middle East and Central Asia”, Comparative Studies in Society and History, Vol. 28, No. 2 (1986) pp. 334-355.

Statistically speaking, EVERY Muslim is a direct descended of Muhammad.

Lets say, for simplicity's sake that Muhammad had only 2 children around the year 600 AD. If a generation lasts 20 years, and each generation has two offspring, we get 70 generations by 2000. This means that Muhammad would have 2^70 related descendants, which equals around 1,000,000,000,000,000,000,000.

/u/Georgy_K_Zhukov goes into way more depth with this in an excellent post about the validity of Ancestry.com using Charlemagne as an example:

http://www.reddit.com/r/AskHistorians/comments/20pubn/how_reliable_is_ancestrycom_is_it_based_on/

Not a historian but I do know that there is an actively maintained family tree for the prophet's bloodline. People who belong to that bloodline are called "Ashraaf (plural)" and they usually have a sort of "union" in almost every Muslim-majority country; though they may differ in how they define this bloodline. In most countries one is only considered "Shareef (singular)" if they would be connected only to the family of Ali and Fatima, and only through the fathers; in Egypt, connections through the mothers are also considered, and recently, they also included those who descend from Zeinab.

Shiias place the "Ashraaf" on high regard; Imams, like Khomeini or the Aga Khan for example, can only be descendants of Ali. Sunnis on the other hand are largely indifferent towards them; I have two friends, and had one boss that are Ashraaf, and it isn't a subject that would frequent our conversations.

The most notable Sunni Ashraaf that I could think of off the top of my head would be the Jordanian Royal family.