evolution - Are there genes in humans from the common ancestor of all organisms?

I very much agree with bitwise's answer. I just want to point out that even in terms of nucleotide sequence there are some extremely conserved genes.

The most highly conserved are ribosomal RNA genes. The image below shows the conservation of 16S rRNAs from archaea, bacteria, and eukaryotes (eukaryotes do not have 16S in their cellular genome but the gene is present in the DNA of mitochondria and chloroplasts). Residues in red are conserved across all domains of life (image source):

A very nice study from 2008 [1] identified tRNA genes as the next most conserved and ABC transporter nucleotide binding domains (NBDs) after that. The alignment below (taken from the paper) shows the conservation of NBDs from various species. Note that they include eukaryotes, archaea and bacteria:
[2]

If you look at orthologous genes in general, without specifying a specific level of sequence identity, then you can find various genes (notably those coding for the enzymes of the glycolysis pathway as mentioned in the comments) that are conserved across most, if not all, cellular life.

So, to answer your question probably yes. At least, there are genes in humans today that can be found in all or almost all other life forms. Whether there were present in LUCA we cannot know for sure, it does seem likely however given how widespread they are today.

Reference

1. Isenbarger T.A. et al, The most conserved genome segments for life detection on Earth and other planets, Orig Life Evol Biosph. 2008 Dec;38(6):517-33.