I recently completed two serving trays myself and was so happy with the result that I am currently working on 5 more.
The first 2 were from cedar and I went with a 5/8" thickness. I added a
1/4" roundover to both sides and it gave the handles a nice feel and really looks great. The cedar also provided for nice light weight trays. The ones I am currently working on are red oak, and while I left the thickness at 5/8" because I like the look and feel on the originals, the oak is *much* heavier, and if I didn't already have the bottom panels glued up I may be tempted to plane them down to 1/2" or possibly even 3/8". But even so they will be solid. Full dimensions are 13 5/8" x 18 1/2". 2 1/2" H on long sides / 4" on handle sides. The width was sized to easily accomodate a common dinner plate.You can take a look at mine at the site listed in my sig line. Click on "Project Gallery" and its at the bottom. I too made a template for the handles and it made quick work of the handle sides.
Good luck.