I'm glad I tried the diet myself, it made a big difference in how my digestion works. And I lost a load of bloating weight from my stomach when I did it, always a bonus But I have to note that it didn't seem to affect my anxiety levels at all.
When I was tested for coeliac disease and it came up negative I still had a chat with a nutritionist at the hospital about how I felt wheat affected me and she talked to me about the gluten free diet - it was well worth a chat with her before starting.