Using the per capita Violent Crime Rate for 2009 and the current Brady Violence Policy Center's rating for states (scores, not stars or rankings) which the highest rankings go to the most restrictive states...
The correlation between violent crime rate and VPC rating is r = 0.04.
This means--essentially--there is no relationship between violent crime rates (per capita) and restrictiveness on guns in general including assault weapons.
What does--interestingly enough--have a decent correlation is the violent crime rate when you run it against state-level data from census.gov is the percentage of black people (r = 0.69), the percentage of white people (r = -0.58), Federal Expenditures per capita (r = 0.63), Population per square mile (r = 0.64), and the property crime rate per capita (r = 0.63).
If you run it the other way--correlating the gun restrictions as embraced by VPC--the highest correlations I found are:
The Gallup poll on Religiosity (Percentage saying "I'm religious", r = 0.33), the ARIS poll on religion's percentage of non-Catholic Christians (r = 0.35), Southern states that participated in the civil war (i.e. former slave states, r = 0.32), and the percentage of black people again (r = 0.32).
If you take that all together, it says Violent crime is more likely to show up in densely populated areas with a high proportion of black people and the laws restricting firearms show up where you have more religious people, more black people, and a history of civil rights violations as well...
(And--yes, I'm highly aware correlation does not equal causation--but the connections between crime rates--race--population density--religion--and restrictive gun laws is something that should really be pulled apart better...)