Another option, for those of you who still stumble upon this post like I did, is to use find -exec:
find . -type f -name '*.txt' -exec cat {} + >> output.file
In my case, I needed a more robust option that would look through multiple subdirectories so I chose to use find. Breaking it down:
find .
Look within the current working directory.
-type f
Only interested in files, not directories, etc.
-name '*.txt'
Whittle down the result set by name
-exec cat {} +
Execute the cat command for each result. "+" means only 1 instance of cat is spawned (thx @gniourf_gniourf)
>> output.file
As explained in other answers, append the cat-ed contents to the end of an output file.