Convert a dta file to csv without Stata software

The frankly-incredible data-analysis library for Python called Pandas has a function to read Stata files.

After installing Pandas you can just do:

>>> import pandas as pd
>>> data = pd.io.stata.read_stata('my_stata_file.dta')
>>> data.to_csv('my_stata_file.csv')

Amazing!

You could try doing it through R:

For Stata <= 15 you can use the haven package to read the dataset and then you simply write it to external CSV file:

library(haven)
yourData = read_dta("path/to/file")
write.csv(yourData, file = "yourStataFile.csv")

Alternatively, visit the link pointed by huntaub in a comment below.

For Stata <= 12 datasets foreign package can also be used

library(foreign)
yourData <- read.dta("yourStataFile.dta")

You can do it in StatTransfer, R or perl (as mentioned by others), but StatTransfer costs $$$ and R/Perl have a learning curve.
There is a free, menu-driven stats program from AM Statistical Software that can open and convert Stata .dta from all versions of Stata, see:

http://am.air.org/