Here are some usage statistics from Quantcast, which is used to track usage of SE sites (you'll see it as a tracking pixel when you check the loaded resources for each page).
Unfortunately, I haven't found how to get better resolution than day.
Using variants of @DaveDuPlantis' query, I created the following charts, that display the requested information in some more detail than the answers provided so far:
All posts on SU by hour of day:
select
datepart(hour, Posts.CreationDate) Hour,
count(Posts.Id) Questions
from Posts
group by
datepart(hour, Posts.CreationDate)
order by Hour asc
All posts of SU by hours of day and day of week:
select
datename(weekday, Posts.CreationDate) Weekday,
datepart(hour, Posts.CreationDate) Hour,
count(Posts.Id) Questions
from Posts
group by datename(weekday, Posts.CreationDate),
datepart(hour, Posts.CreationDate)
order by Weekday, Hour
I manually fixed the output to bring order to the weekdays
Charts for 2011 only (using Oct 4 data set)
The following are the same charts, but limited to posts created in 2011 only, by adding the following to the queries above:
where YEAR(CreationDate) = 2011
Both charts for 2011 only are basically identical to those for the whole lifetime of the site. The absolute numbers are way down though, so it's not the domination of 2011 in the first charts that causes this: It's just that there was no shift in the times people post to the site.