Seedlings from cultivars may retain some of the characteristics of the parent cultivar, but it can't be relied on.

Resin

PS . . .

Here is a link that might be useful: http://en.wikipedia.org/wiki/Specie ;-)

They may go back three generations or to the mother plant

I believe that in the horticultural trade that cultivars are vegetatively propagated "sports" (an oddball plant that is atypical of its geneal genotype and that has desirable marketable traits). Vegetative propagation of such are generally prohibited without a license from the source of the cultivar. Seedling propagation is not a reliable source of carrying on the cultivar as the genotype is no longer identical to the phenotype represented by the cultivar.

I do know that the local Blacberry Himalaya comes dreadfully true from seed. It will cross with Evergreen when they are near one another and the hybrids can be actually better than either parent.
A lot of citrus sets a form of seed which is genetically identical to the parent, but I cannot recall what this is called in Latin.
Some tree fruits are very much like their parent from seed-such as Italian Plum, but Apples and Grapes are very different usually.

Here is a plant that produces identical seedlings to the thorny parent: Hardly recommend it though.

Reproduction
Himalayan blackberry reproduces by means of extensive asexual reproduction via numerous methods and less so by sexual reproduction. The asexual means include: 1) apomixis with segregation, 2) apomixis without segregation, 3) haploid parthenogenesis, and 4) non-reduction at meiosis (Tirmenstein 1989).
Vegetative regeneration means include: 1) re-growth from the perennial rootstalk; 2) rooting stem tips (Tirmenstein 1989) and 3) root and cane fragments (Alaska Natural Heritage Program 2005).
The extensive asexual reproduction was proven when genetic testing in Australia indicated that a majority of the population in southern Australia was from a single clone. The DNA fingerprinting did not show any genetic variation from the 29 sampled locations (Evans et al. 1998).

OK, Resin's link was apparently not the one he meant to post. Maybe he would have directed you to the page I will link below. I only skimmed it quickly, but I think it gave a pretty good explanation of the basic science behind this question.

A couple of points about terrestrial_man's post above....Most cultivars are not patented and therefore would not require any type of license to propagate. Also, many cultivars (cultivated varieties) are sexually (by seed) reproduced. Again, Wikipedia probably explains sexually v. asexually produced cultivars in the following article:
http://en.wikipedia.org/wiki/Cultivar

Here is a link that might be useful: Wikipedia Article on Genes

"OK, Resin's link was apparently not the one he meant to post"

It was the right link . . . I was pointing out that specie (see the thread title!) is NOT the same as species - the former is coins, the latter is types of living things ;-)

Resin