Here is one-line list-comprehension variant of Pat's answer (which also includes that you wanted to glob in a specific project directory):
import os, glob
exts = ['*.txt', '*.mdown', '*.markdown']
files = [f for ext in exts for f in glob.glob(os.path.join(project_dir, ext))]
You loop over the extensions (for ext in exts), and then for each extension you take each file matching the glob pattern (for f in glob.glob(os.path.join(project_dir, ext)).
This solution is short, and without any unnecessary for-loops, nested list-comprehensions, or functions to clutter the code. Just pure, expressive, pythonic Zen.
This solution allows you to have a custom list of exts that can be changed without having to update your code. (This is always a good practice!)
The list-comprehension is the same used in Laurent's solution (which I've voted for). But I would argue that it is usually unnecessary to factor out a single line to a separate function, which is why I'm providing this as an alternative solution.
Bonus:
If you need to search not just a single directory, but also all sub-directories, you can pass recursive=True and use the multi-directory glob symbol ** 1:
files = [f for ext in exts
for f in glob.glob(os.path.join(project_dir, '**', ext), recursive=True)]
This will invoke glob.glob('<project_dir>/**/*.txt', recursive=True) and so on for each extension.
1 Technically, the ** glob symbol simply matches one or more characters including forward-slash / (unlike the singular * glob symbol). In practice, you just need to remember that as long as you surround ** with forward slashes (path separators), it matches zero or more directories.