Do people in mourning behave differently according to their culture?
There is enough research that amply points at mourning as a cross-cultural and complex set of behaviors that are expected to take place during bereavement.
The reason is because of the different family-rearing dynamics among the cultures, which in turn cause different levels of connection. This, added to the social and individual expectations set by society upon family members of different cultural groups, is partly why the process of bereavement and mourning is so different and complex.
Take for example the universally-known rendition to death and family known as "The Day of the Dead" or "Día de los Muertos" in Mexico. Far from a sad celebration, the Day of the Dead celebrates life despite of its name. The dead are put in a center stage and their favorite things are placed on an altar which is decorated with mums and other mementos of the deceased. The dedication, excitement and detail placed upon this celebration contrasts dramatically with some variations of Afro-Caribbean families, who openly show their grief with loud expressions of pain as a way to release their emotions and at the same time to create awareness of the absence of the deceased. This does not prescribe the same attitude towards death from all the other variations of Hispanics; the temperament, cause of loss, age of the deceased, and the personal connection to the deceased are also factors in every culture, but Dia de los Muertos offers a rare glimpse into the psyche of the Mexican Culture.
Another example is the Amish community, which goes on full mourning for a year after the passing away. The Amish, in an interestingly similar behavior to some Asian cultures, prefer to conduct their mourning in private and public displays of emotion are not common. Regardless, this has nothing to do with personal emotion; it is merely the way in which they are chosen to be displayed.
Finally, although the list could go on, the Chinese, as a complex culture that is rife with symbolism and ceremony, express their grief by "sending away" their beloved in a combination of symbolic burnt offerings that aims to internalize grief. The different ceremonies correspond to the manner in which this internalization intends to make sense out of the loss and apply to it some psychological meaning.
A wonderful source to consult for further information is the National Institutes for Health and the U.S. National Library of Medicine. They offer a number of resources available to the public with articles written exclusively about the cross-cultural differences of the grieving process.